By Math Vault | Algebra

For the very vast majority of humans on earth, there is a topic found in the good old math textbooks that many of us still even dread contemplating about, as it seems to mess with our brain in a rather* particular* way. The name? $\displaystyle \text{Logarithmus}$ — or **Logarithm** in English to be sure!

As terrible-sounding as it is, logarithm seems to have this *distinct* characteristic of metaphorically leaving a *bad taste* in our mouth. In fact, even for those who managed to maneuver around it back in high school, logarithm still remains largely as an *evasive* concept. The **“I-can-manipulate-expressions-without-understanding-anything” syndrome** runs rampant when it comes to logarithm.

Indeed, here in North America, the grade school curriculum has the propensity of overemphasizing the **mechanics** at the expense of **basic theory**, leaving us with the *formidable* task of filling in the *logarithmic* knowledge gap, which includes — among others — the theory behind the **properties of logarithm**, and its intended **computational use** in handling numbers with an **order of magnitude** veering towards the extremes.

So with that in mind, if you think that the time might have finally come to tame this *monster* we call logarithm, then it would be our pleasure to congratulate your timing on this very honorable act. And if you are simply looking to explore further into the rabbit hole, that would be *doubly* appreciated as well, for regardless of your motivation, the **taming**/**musing** is on! 🙂

Given a *real number* $x$, one of the challenges in **elementary algebra** is to express $x$ as a *power* of another number $b$ (known as the **base**). More specifically, we are interested in finding a number $\Box$ such that:

\begin{equation*} x = b^\Box \end{equation*}

As it turns out, this problem — in the *crude* form that it currently is at least — needs to be patched up first before any meaningful discussion can take place. For example:

- If the base is
*negative*, then its powers need not be necessarily**well-defined**(e.g., $\displaystyle (-e)^{\frac{1}{2}}$). - If the base is $\displaystyle 1$, then any power of it would be just $1$, in which case, it would be
*impossible*for it to generate any number that’s not $1$. A similar remark applies to the case where the base is equal to $0$.

For these reasons, in the context of **power determination**, it’s customary to require the base $b$ to be a *positive* number — that is not equal to $1$. While under this assumption, any power of $b$ would necessarily have to be *positive*, it would also transpire —under this setup — that *any* positive number can be expressed as a power of $b$ in a *unique* way. That is, as long as $x$ is *positive*, there will be a *unique* number $\Box$ (known as the **exponent**) such that:

\begin{equation*} x= b^{\Box} \end{equation*}

in which case, we will simply call $\Box$ the **logarithm** of $x$ (in base $b$). In other words, logarithm is basically what happens when we expressed a number as a *power*, and then take the *exponent* from that power — It gives us the **magnitude** of a number, with respect to the base in question.

For example, when we try to express the number $64$ as a power of $2$, we get that $64= 2^6$. This alone shows that $6$ is the logarithm of $64$ — with respect to the base $2$.

Notation-wise, the logarithm of $x$ in **base** $b$ is denoted by $\log_b x$, with $x$ also being called the **argument** of the logarithm. When considered as a function, $\log_b x$ is defined on all *positive* numbers — as long as the base $b$ is **valid** (i.e., $\displaystyle b>0, b \ne 1$) .

To begin, we first note that *regardless* of the value of the base $b$, we always have that:

- $\displaystyle \log_b 1 = 0$ (since $0$ is the number $b$ needs to be raised to yield $1$)
- $\displaystyle \log_b b = 1$ (since $1$ is the number $b$ needs to be raised to yield $b$)
- $\displaystyle \log_b \frac{1}{b} = -1$ (since $-1$ is the number $b$ needs to be raised to yield $\displaystyle \frac{1}{b}$)

Because these results are almost immediate and sufficiently notable, we’ll simply refer to them as the **trivial logarithmic identities**.

In addition, since $\log_b x$ stands for the number which *exponentiates* to $x$, we also have that by definition:

\begin{align*}b^{\log_b x} & = x \qquad (\text{for all }x>0)\end{align*}

On the other hand, we also have that:

\begin{align*} \log_b (b^x) = x \qquad (\text{for all } x \in \mathbb{R}) \end{align*}

Since one can see by inspection that $x$ is precisely the number which exponentiates to $b^x$.

For example, since $\displaystyle \log_2 53$ is the number that $2$ needs to raise to yield $53$, we have that $\displaystyle 2^{\log_2 53} =53$. Similarly, since $\displaystyle 10^{-\pi}$ is a power of $10$ with the exponent $-\pi$, we can infer that $\displaystyle \log_{10} \left(10^{-\pi}\right) = -\pi$.

Being the inverse of the exponential function $\displaystyle 10^x$, the base-$10$ logarithmic function — also known as the **common logarithm** — is customarily denoted by $\log_{10} x$, $\log x$, or simply $\lg x$ for short. The common logarithm is of great interest to us, primarily due to the prevalence of the **decimal number system** in various cultures around the world.

When the common logarithm of a number is calculated, the *decimal representation* of the logarithm is usually split into two parts: the integer component (a.k.a., **characteristic**) and the fractional component (a.k.a., **mantissa**). The characteristic in essence tells us the **number of digits** the original number has, and the mantissa hints at the extent to which this number is close to its next power of $10$. These are the facts that make common logarithm a particularly handy tool in determining the **order of magnitude** of an *exceptionally large* (or *small*) number.

For example, to figure out the magnitude of the number $50!$ (i.e., $50 \times \cdots \times 1$), we proceed to calculate its logarithm, yielding that $\log (50!) \approx 64.483$. Translation? $50! \approx 10^{64.483} = 10^{64}10^{0.483} \approx 10^{64} \cdot 3.04$, suggesting that $50!$ is a $65$*-digit number* which starts with $3$ — The **characteristic** $64$ gives away the number of digits, and the **mantissa** $0.483$ reveals the rest about the number itself.

Take home message? There is no need to write out a number in full to figure out its *approximate size*!

Being the inverse of the exponential function $2^x$, the **binary logarithm** function $\log_2 x$ is extensively used in the field of **computer science**, primarily due to the fact that computers store information in **bits** (i.e., digits which takes $0$ or $1$ as possible values).

Similar to the case in base $10$, binary logarithm can be used to figure out the number of digits of a positive integer in **binary representation**. In addition, binary logarithm is also used to figure out the *depth* of a **binary tree**, or even the *number of operations* required by certain **computer algorithms** (this falls into a topic known as **algorithmic time complexity**).

Beyond the world of computers, binary logarithm is also used in **music theory** to conceptualize the *highness* of musical notes, based on the fundamental observation that *raising* a note by an **octave** increases the frequency of the note by *twofold*. As a result, it is often convenient to conceive a **musical interval** as the binary logarithm of the **frequency ratio**.

In some textbooks concerned with a more rigorous development of **transcendental functions**, the base-$\displaystyle e$ logarithmic function — otherwise known as **natural logarithm**, $\log_e x$ or simply $\ln x$ — are sometimes defined as the *area* between the **reciprocal function** $\frac{1}{x}$ and the x-axis from $1$ to $x$ (hence the term *natural*).

Under this definition, it could be shown that the inverse of $\ln x$ is precisely the **natural exponential function** $e^x$, leading to the following *standard* definition of natural logarithm:

## Given a positive number $x$, $\ln x$ denotes the number that $e$ needs to be raised, to become $x$.

Unlike the number $10$ — which is preferred due to the prevalence of **decimal numbering system** — the number $\displaystyle e$ is one special* constant* that pops up surprisingly often in various mathematical discourses — *irrespective* of the number system being chosen. As a result, mathematicians tend to consider base $e$ as more *natural* than base $10$ — even though some applied scientists and engineers beg to differ in various occasions…

Actually, to illustrate the scope of these *intellectual biases* among the scientific community, here’s an interesting account from Wikipedia on the **historical development** of the notations for logarithms:

## Because base 10 logarithms were most useful for computations, engineers generally simply wrote “log(x)” when they meant log

_{10}(x). Mathematicians, on the other hand, wrote “log(x)” when they meant log_{e}(x) for the natural logarithm. Today, both notations are found. Since hand-held electronic calculators are designed by engineers rather than mathematicians, it became customary that they follow engineers’ notation. So the notation, according to which one writes “ln(x)” when the natural logarithm is intended, may have been further popularized by the very invention that made the use of “common logarithms” far less common, electronic calculators.

In addition to the three most popular logarithmic functions introduced earlier, one can also define logarithm using other **valid bases** as well. In practice, logarithm is generally employed with the intention of *condensing* large numbers (i.e., greater than $1$) into smaller numbers, so that the *larger* the base, the *smaller* the logarithm.

However, that’s only part of the story, as all the logarithmic functions we have encountered so far have bases exceeding the number $1$ (i.e., **large** **base**). In fact, in the cases where the base is *strictly* between $0$ and $1$ (i.e., **small** **base**), the graph of the logarithmic function will be turned *upside down*. Indeed, as opposed to a standard logarithmic function which *increases* from $-\infty$ to $\infty$, a small-base logarithmic function actually *decreases* from $+\infty$ to $-\infty$ as the argument increases.

Fortunately though, we rarely have to resort to this kind of logarithmic function in practice. As we shall see later with the **Change of Base Rule**, every pair of logarithmic functions are all but a *multiple* apart, so that in terms of **applications** and **equation/inequality solving**, the three standard logarithms are.generally more than enough to get things going.

Since logarithm allows for mapping an exponential scale into a **linear scale**, it became an vital concept when it comes to communicating about a numerical variable whose quantities either *grow* exponentially, or *shrink* exponentially. Why? Because if we just take the logarithm of that variable, we are in effect turning that variable into something people refer to as a **logarithmic scale**.

While seemingly a highly-theoretical concept, logarithmic scale — when adopted* appropriately* — can be used to help us better explain/understand a surprising amount of phenomena found in *nature*. ranging from stuffs such as the **loudness of a sound**, the **magnitude of an earthquake** to the **acidity of a solution** and the **highness of a musical pitch**:

- In an
**equal-tempered piano**, each key in the piano can be conceived as the*binary**logarithm*of its*relative*sound frequency, so that every time we press a higher key on the piano, we are in effect increasing the sound frequency by a fixed factor ($\displaystyle \sqrt[12]{2}$ to be precise). - In
**chemistry**, the acidity of a solution is measured in**pH**, which is defined as the*negative logarithm*of the**concentration of hydrogen ions**. This basically means that as pH increases by $1$, the concentration of hydrogen ions decreases*tenfold*, leading to a substantially less acidic solution in return. - In
**seismology**, the severity of an earthquake can be quantified using the**Richter scale**, which is essentially the*logarithm*of the**amplitude of seismic waves**(relative to a*threshold amplitude*), so that every time the number on Richter scale increases by $1$, severity of the earthquake increases*tenfold*. - In
**acoustics**, the loudness of a sound is generally quantified using**decibel**(dB), which is a tenth of a**bel**(B), the latter of which is the*logarithm*of the**sound power**(relative to the*threshold of hearing*). In practice, this means that every increase of 10 dB (equiv., 1B), increases the sound power of a source by*tenfold*.

While not all logarithmic scales share the same base, the fact that one can hijack a concept such as logarithm — originally a *purely-computational* device — into an **unifying abstract framework** stringing together various seemingly unrelated phenomena found in nature, illustrates the power of the human mind and by extension, the thin line between **mathematical invention** and **discovery**.

One of the reasons why logarithm was such a powerful *computational* tool back in the old days — before the invention of *computers* or *calculators* — lies in the fact that one can always leverage certain **properties of logarithm** to reduce a *complicated argument *to its *individual constituents *— and doing so irrespective of the base in question. In what follows, we lay out *five* of such properties, which pertains to the **product**, the **reciprocal**, the **quotient**, the **power** and the **root** of a logarithm, respectively.

Given a product $xy$ and a base $b$, can we find the logarithm of $xy$ in terms of the logarithms of $x$ and $y$? As it turns out, the answer is a resounding *yes*, and a bit of inspection shows that $\log x + \log y$ is the number we are looking for. How so? Because that’s the number $b$ needs to be raised to get to $xy$:

\begin{align*} b^{\log x + \log y}= b^{\log x} \cdot b^{\log y} = xy \end{align*}

This fact — that logarithm of a product can be reduced into sum of logarithms of its constituents — gives rise to a property commonly known as the **Product Rule**.

Rule 1 — Product Rule for Logarithm

Given any two *positive* numbers $x$, $y$, we have that:

\begin{align*} \log (xy) =\log x + \log y \end{align*}

where all logarithms are assumed to be under the same *valid* base $b$.

In particular, when the base is $10$, the Product Rule can be translated into the following statement:

## The magnitude of a product, is equal to the sum of its individual magnitudes.

For example, to gauge the approximate size of numbers like $365435 \cdot 43223$, we could take the *common* logarithm, and then apply the **Product Rule**, yielding that:

\begin{align*} \log (365435 \cdot 43223) & = \log 365435 + \log 43223 \\ & \approx 5.56 + 4.63 \\ & = 10.19 \end{align*}

which shows that $\displaystyle 365435 \cdot 43223$ is a *11-digit* number close to $\displaystyle 10^{10.19} \approx 1.55 (10^{10})$.

Only apply the Product Rule when the preconditions are met. For example, one thing we cannot do is to break down $9$ into $-1$ and $-9$, and claim that $\ln 9 = \ln(-1) + \ln (-9)$.

And because an equality is by default *bidirectional*, instead of *breaking the product* by using the Product Rule *from the left to the right*, we can also use it *from the right to the left*, thereby turning a sum of logarithms into a product instead. For example:

\begin{align*} \log 25 + \log 4 = \log (25 \cdot 4) = \log 100 = 2 \end{align*}

On the downside however, since logarithm by default takes only *positive* numbers as arguments, applying logarithm and its properties to a *function* or an *equation* can significantly restrict its **domain of feasibility**. For example, while the function $x^2$ is defined on all *real* numbers, once we take the logarithm and apply the *Product Rule*, the resulting equality is still applicable — but now to the *positive* numbers *only*:

\begin{align*} \log (x^2) = \log x + \log x = 2 \log x\end{align*}

which serves as a good reminder that any *logarithm-based* algebraic technique — be it **logarithmic equation solving** or **logarithmic differentiation** — should be carried out with this potential restriction in mind.

We know that every positive number has a multiplicative inverse (i.e., **reciprocal**), so perhaps there is also a *shortcut* in finding the logarithm of a reciprocal? Here again, the answer is a resounding *yes*. To see why, suppose that we are are given a positive number $x$, then by **Product Rule**, we have that:

\begin{align*} \log \left(x \ \cdot \frac{1}{x} \right) & = \log x + \log \left( \frac{1}{x} \right) \end{align*}

On the other hand, we also have that:

\begin{align*} \log \left(x \cdot \frac{1}{x} \right) & = \log 1 = 0 \end{align*}

*Bridging* the two equations together, we get that:

\begin{align*} \log x + \log \left( \frac{1}{x} \right) & = 0 \end{align*}

Or equivalently,

\begin{align*} \log \left( \frac{1}{x} \right) & = – \log x \end{align*}

Surprise! We just discover the **Reciprocal Rule**, which states that to find the logarithm of a reciprocal, we just have to *negate* the logarithm of the original number.

Rule 2 — Reciprocal Rule for Logarithm

Given any *positive* numbers $x$, we have that:

\begin{align*} \log \left(\frac{1}{x} \right) = – \log x \end{align*}

where all logarithms are assumed to be under the same *valid* base $b$.

So instead of calculating the *binary* logarithm of $\displaystyle \frac{1}{512}$ from scratch, we could *turn its head around* and do:

\begin{align*} \log_2 \left( \frac{1}{512} \right) = – \log_2 512 = – 9 \end{align*}

Now that both **Product Rule** and **Reciprocal Rule** are in order, let’s see what happens if we apply them to a **quotient** of *positive* numbers $x$ and $y$:

\begin{align*} \log \left( \frac{x}{y} \right) & = \log \left( x \cdot \frac{1}{y} \right) \\ & = \log x + \log \left( \frac{1}{y} \right) \\ & = \log x – \log y \end{align*}

Bingo! We have just shown that the logarithm of a quotient is precisely the *difference* between the original logarithms — a property commonly known as the **Quotient Rule**.

Rule 3 — Quotient Rule for Logarithm

Given any two *positive* numbers $x$, $y$, we have that:

\begin{align*} \log \left( \frac{x}{y} \right) =\log x – \log y \end{align*}

where all logarithms are assumed to be under the same *valid* base $b$.

For example, instead to computing the *natural logarithm* of $\displaystyle \frac{2}{e}$ from scratch, we could apply the *Quotient Rule*, and get that:

\begin{align*} \ln \left( \frac{2}{e}\right) = \ln 2 – \ln e = \ln 2 – 1 \end{align*}

As in the case with **Power Rule**, instead of *breaking the quotient*, we can also use the Quotient Rule *from the right to the left*, thereby turning a *difference* into a *quotient* instead. For example:

\begin{align*} \log 45 – \log 9 = \log \left( \frac{45}{9} \right) = \log 5 \end{align*}

In base $10$, the Quotient Rule can also be translated into the following insight:

## The magnitude of a quotient, is equal to the difference of the individual magnitudes.

which explains why in natural science, a quantity is often expressed in **logarithmic scale**,** **by taking the logarithm of the **ratio** between the *said quantity* and a *reference point*.

As for the logarithm of a number raised to an *integer* power, we begin by noting the case where a number is raised to $0$:

\begin{align*} \log (x^0) = \log 1 = 0 = 0 \log x \end{align*}

In the case where a number is raised to a *positive* integer $n$, the logarithm can be obtained through the* repeated applications* of **Product Rule**:

\begin{align*} \log (x^n) & = \log \underbrace{ \left(x \cdots x \right)}_{n \text{ times}} \\ & = \underbrace{\log x + \, \dots + \log x}_{n \text{ times}} \\ & = n \log x \end{align*}

And in the case where a number is raised to a *negative* integer of the form $-n$, a mix of **Product Rule** and **Reciprocal Rule** will do:

\begin{align*} \log (x^{-n}) & = \log \left[ \left( \frac{1}{x}\right)^n \right] \\ & = n \log \left( \frac{1}{x}\right) \\ & = -n \log x \end{align*}

Either way, we’ve just shown that when a number is raised to a *integer* power, the resulting logarithm is *rescaled* precisely by that power as well. This interesting finding would result in a *key* property of logarithm known as the** Power Rule**.

Rule 4 — Power Rule for Logarithm

Given any *positive* numbers $x$ and a integer $n$, we have that:

\begin{align*} \log (x^n) = n \log x \end{align*}

where all logarithms are assumed to be under the same *valid* base $b$.

As some might have expected, the **Power Rule** is by itself a very *powerful* property. For one, it allows us to *pull out* the exponent from the argument of a logarithm, thereby *normalizing *a potentially *gigantic */ *minuscule* number (e.g., $\displaystyle \log_2 (3^{15}) = 15 \log_2 3$), Conversely, the Power Rule can also be used to *push* an exponent inside the argument of a logarithm, thereby producing a *potentially-simpler* expression (e.g., $\displaystyle 3 \ln 5 = \ln (5^3) = \ln 125$).

Make sure that the precondition is met before applying the Power Rule. For example, while $\ln (x^8)$ is defined on all *non-zero* numbers, the equation $\ln (x^8)=8 \ln x$ is only true when $x>0$. In this case though, the issue can be resolved by *absolutizing* $x$, yielding the equality $\ln (|x|^8) = 8 \ln |x|$ instead.

Similar to the case with **Product Rule** and **Quotient Rule**, Power Rule can be interpreted as follows in base $10$:

When a number is powered, the magnitude of the resulting number is precisely equal to the magnitude of the original number — times that power.

To find some *shortcut* in evaluating the logarithm of a *root*, we begin by observing that for all *positive* integer $n$, an application of** Power Rule **shows that:

\begin{align*} \log \left[ (\sqrt[n]{x})^n \right] = n \log (\sqrt[n]{x}) \end{align*}

On the other hand, we also have that:

\begin{align*} \log \left[ (\sqrt[n]{x})^n \right] = \log x \end{align*}

*Bridging* the two equalities together, we get that:

\begin{align*} n \log (\sqrt[n]{x}) = \log x \end{align*}

Or equivalently,

\begin{align*} \log (\sqrt[n]{x}) = \frac{\log x}{n} \end{align*}

Awesome! This shows that to figure out the logarithm of a $n$^{th} root, all we have to do is to divide the logarithm of the original number by $n$ — An insight which results in another property of logarithm known as the **Root Rule**:

Rule 5— Root Rule for Logarithm

Given any *positive* number $x$ and a positive integer $n$, we have that:

\begin{align*} \log (\sqrt[n]{x}) = \frac{\log x}{n} \end{align*}

where all logarithms are assumed to be under the same *valid* base $b$.

Much like the **Power Rule**, the Root Rule is not only useful for its ability to *pull* out the root from the logarithm (as in $\displaystyle \log (\sqrt[12]{6}) = \frac{\log 6}{12}$), but for its ability to *create* a root out of nothing as well (as in $\displaystyle \frac{\ln 2}{5}=\ln (\sqrt[5]{2})$).

In base $10$, the Root Rule can be interpreted as follows:

## When a number is rooted, the resulting magnitude is rescaled precisely by the degree of the root in question.

And when we combine **Power Rule** and **Root Rule** together, we get that for any *rational number* of the form $\displaystyle \frac{m}{n}$ ($m \in \mathbb{Z}, n \in \mathbb{N}$):

\begin{align*} \log x^{\frac{m}{n}} & = \log \left[ (\sqrt[n]{x})^m \right] \\ & = m \log (\sqrt[n] x) \\ & = \frac{m}{n} \log x \end{align*}

In fact, this is nothing more than a *special instance* of the **Generalized Power Rule**:

\begin{align*} \log (x^p) = p \log x \qquad (\text{for all }p \in \mathbb{R}) \end{align*}

which can be proved by showing that $p \log x$ is indeed the *exponent* to which the base $b$ needs to be raised — to produce $x^p$:

\begin{align*} b^{p \log x} & = \left( b^{\log x }\right)^p = x^p\end{align*}

And finally, here is an example illustrating *all* the *argument-related* properties of logarithm we have seen thus far:

\begin{align*} \log \left( \frac{5^3 \cdot \sqrt[4]{15}} {10^{66} \cdot e^{\pi}} \right) & = \log \left( 5^3 \cdot \sqrt[4]{15} \right) – \log \left( 10^{66} \cdot e^{\pi} \right) \\ & = \left [\log 5^3 + \log \sqrt[4]{15} \right] – \left[ \log (10^{66}) + \log (e^{\pi})\right] \\ & = 3 \log 5 + \frac{\log 15}{4} – 66 \log 10 – \pi \log e \end{align*}

OK. So that was a bit of properties involving the **argument** of logarithm, but what if we want to tweak with the **base** instead? Well, we’ve got you covered on that one too! In what follows, we present *four* properties of logarithm involving the bases for your own pleasure. THese are the **Chain Rule**, the **Change-of-Base Rule**, the **Base-Swapping Rule** and the **Base-Argument Interchangeability**.

Suppose for a moment that you have mastered all the *five* properties of logarithm used to reduce a argument into its *simplest* constituents, but for one reason or another find the **base** rather annoying. What would you do?

In our case, we would to find an alternate formula for calculating the same logarithm — without having to resort to this base *directly*.

Actually, let’s begin by *tightening* the question a bit: given a *positive* number $x$, is there a way of calculating $\log x$ (under base $b$) using a *new* base $a$? Or even better: what’s an *expression* involving $a$, such that when $b$ raised to it, becomes $x$?

Here, to find one such expression, it would be just natural to start with $\displaystyle \log_a x$ and see where it takes us from there. By definition, $\displaystyle \log_a x$ is just the number that $a$ needs to be raised, to become $x$. That is:

\begin{align*} a^{\log_a x} = x \end{align*}

But herein lies a problem: we want the left hand side to be a power of $b$ though. How can we turn that $a$ into $b$? Well, if we write $a$ as a *power of $b$* that is! With this newfound insight, we proceed to replace the bottom $a$ with $\displaystyle b^{\log a}$, yielding that:

\begin{align*} {\left( b^{\log a} \right)}^{\log_a x} = b^{\log a \cdot \log_a x} = x \end{align*}

Bingo! The exponent in the middle term is exactly what we were looking for — the expression that $b$ needs to be raised, to become $x$! We can therefore conclude that:

\begin{align*} \log x = \log a \cdot \log_a x \end{align*}

In English, this reads:

## The logarithm of a number, can be evaluated as the logarithm of a

newbase,times the logarithm of the original number under that new base.

In fact, this property is so impressive, that we decided to *baptize* it as the **Chain Rule** (why?).

Rule 6— Chain Rule for Logarithm

Given any *positive* numbers $x$ and a *new* valid base $a$, we have that:

\begin{align*} \log x = \log a \cdot \log_a x \end{align*}

where all the logarithms whose base isn’t explicitly defined are assumed to be under the same *valid* base $b$.

Here, an example illustrating its use is definitely called for: suppose that we’re given the task of determining the **magnitude** of the number $1024$ (i.e., find $\log_{10} 1024$), but figure that it would easier if the base were in $2$ instead, then one thing that we can do would be to apply the Chain Rule with $2$ as the *new base*, yielding that:

\begin{align*} \log_{10} 1024 = \log_{10} 2 \cdot \log_2 1024 = \log_{10} 2 \cdot 10 \approx 3.01 \end{align*}

so the magnitude of $1024$ is approximately $3.01$ (i.e., $1024 = 10^{3.01}$), which is consistent with it being a *4-digit* number.

In fact, here is more: it actually doesn’t matter which new base we choose to use! The base could have been $3$, $e$ or even $\pi$, and the result would have been exactly the same!

\begin{align*} \log_{10} 1024 & = \log_{10} 3 \cdot \log_3 1024 \\ & = \log_{10} e \cdot \log_e 1024 \\ & = \log_{10} \pi \cdot \log_{\pi} 1024 \end{align*}

If you’re still hanging around, you remember that **Chain Rule** states that:

\begin{align*} \log x = \log a \cdot \log_a x \end{align*}

Here, if we just solve for $\log_a x$, we would get:

\begin{align*} \log_a x = \frac{\log x}{\log a} \end{align*}

Goodness! Another *alternate* formula for logarithm! Except that this time, it’s an *(in)famous* one for real. The name? **Change-of-Base Rule**!

Rule 7— Change-of-Base Rule for Logarithm

Given any *positive* numbers $x$ and a *valid* base $a$, we have that:

\begin{align*} \log_a x = \frac{\log x}{\log a} \end{align*}

where the logarithms whose base isn’t explicitly defined are assumed to be under the same *valid* base $b$

At the first sight, this might seem like a mere reformulation of the **Chain Rule**. However, upon further inspection, one can see that this is actually not the case: what the Chain Rule does is to turn a logarithm into a *product* of logarithms with *different* bases, while the **Change-of-Base Rule** turns a logarithm into a *quotient* of logarithms with the *same* base. In addition, the Chain Rule *marginally* facilitates the evaluation of a logarithm under a new base, while the Change-of-Base Rule actually *fully* eliminates the dependence on the old base.

In practice, the **Change-of-Base Rule** is primarily used to compute a “non-standard” logarithm by turning it into a quotient of “standard logarithms” (e.g., $\log_{15} 26 = \frac{\ln 26}{\ln 15} = \frac{\log_2 26}{\log_2 15}$). However, it can also be used in the reverse manner, thereby *merging* a quotient of logarithms into a single logarithm (e.g., $\frac{\ln 8}{\ln 2} = \log_2 8 = 3$).

In fact, both the **Chain Rule** and the **Change-of-Base Rule** provide a unifying framework for logarithms of *all* valid bases, by showing that every logarithmic function is a **multiple** apart from one another. With the advent of *computers* and *calculators*, the Change-of-Base Rule also opens up the practice of calculating a logarithm of an arbitrary base, by *standardising* it to base $e$, or — if a scientific calculator is used — to base $10$.

Now, let’s do something fun. Remember that the **Change-of-Base Rule** states that:

\begin{align*}\log_a x = \frac{\log x}{\log a} \end{align*}

with the understanding that the “baseless” logarithms are actually assumed to be under some *valid* base $b$. Out of curiosity, if we just let this base to be $a$, we get that:

\begin{align*}\log_a x = \frac{\log_a x}{\log_a a} = \log_a x \end{align*}

which is not terribly interesting, as we are basically going in full circle. However, in the very special case where $x \ne 1$, we can let $x$ to be the base instead, thereby producing the following identity:

\begin{align*}\log_a x = \frac{\log_x x}{\log_x a} = \frac{1}{\log_x a}\end{align*}

Impressive! It’s almost like playing **LEGO®**! And since no one has given it a name yet, let’s just jump in and baptize it as the **Base-Swapping Rule**.

Rule 8 — Base-Swapping Rule for Logarithm

Given any two *valid* bases $x$ and $a$, we have that:

\begin{align*}\log_a x = \frac{1}{\log_x a}\end{align*}

In English, the **Base-Swapping Rule** translates into the following insight:

## An alternate way of figuring out the logarithm of a number under a base, is to find the logarithm of the base under that number, and then take the reciprocal.

Terrible pun we know, but if you really *hate* operating under a certain base, the Base-Swapping Rule provides a *quick-and-dirty* way to get rid of it. For example:

\begin{align*} \log_{512} 2 = \frac{1}{\log_2 512} = \frac{1}{9}\end{align*}

which shows that **mental LEGO** can be just as fun as playing with a few dozen pieces of *colored plastic*. 🙂

As a general rule of thumb, we don’t want to mess around with the **bases** and **arguments** by swapping them around. However, in the very special case where a base is raised to a *logarithm*, a bit of swapping actually preserves the equality, and is sometimes even preferred — like this:

\begin{align*} x^{\log y} & = y^{\log x} \end{align*}

In case you’re wondering about the exact nature of this black magic, here’s a proof showing how the left-hand side becomes the right-hand side:

(**Note**: all logarithms are assumed to be under base $b$)

\begin{align*} x^{\log y} & = {\left( b^{\log x} \right)}^{\log y} \\ & = {\left( b^{\log y} \right)}^{\log x} \\ & = y^{\log x} \end{align*}

Impressive property! For the lack of better term, let’s just refer to it as the **Base-Argument Interchangeability**!

Rule 9 — Base-Argument Interchangeability for Exponent

Given any two *positive* numbers $x$ and $y$, we have that:

\begin{align*} x^{\log y} & = y^{\log x} \end{align*}

where all logarithms are assumed to be under the same *valid* base $b$

To illustrate the mechanics of this amazing property, here’s a fancy example for your pleasure:

\begin{align*} {\left( 2^{\ln 3} \right)}^{\ln 4} & = {\left( 3^{\ln 2} \right) }^{\ln 4} & (\text{swapping }2 \text{ and }3)\\ & = 4^{\ln \left( 3^{\ln 2} \right) } & (\text{swapping } 3^{\ln 2} \text{ and } 4)\\ & = 4^{\ln \left( 2^{\ln 3} \right) } & (\text{swapping }3 \text{ and }2)\\ & = {\left( 2^{\ln 3} \right)}^{\ln 4} & (\text{swapping }4 \text{ and }2^{\ln 3})\end{align*}

Kind of fun, right? Great place to pull the curtain on the properties of logarithm too! 🙂

As you might have heard people saying on multiple occasions, logarithm — for most practical purposes at least — is defined *only* on the *positive* real numbers. Why? Because a number can only have logarithm if it’s expressible as a *power*, which in turn must be positive — by virtue of the definition of *real-valued* **exponential functions**.

However, as one would expect, this doesn’t sit well with a certain group of *mathematical freedom fighters*, for whom the following question might be more relevant:

## Is there anything we can do to expand the domain of logarithmic functions to other numbers as well?

The answer? Yes, but not without some string attached. As it turns out, *forcing* this domain expansion will inevitably incur some *painful* sacrifices on many fronts, which include — among others — a substantial loss in the properties of **exponent** and **logarithm**.* *

To begin, we do know that defining the **natural exponential function** *solely* on the real numbers is a *major stumbling block* which needs to go, for it is precisely what restricted the exponentials to the positive numbers in the first place. To extend the definition of the natural exponential function to *any* **complex number**, we begin by *redefining* the exponential function as follows:

\begin{align*} e^{x+yi} \, \stackrel{def}{=} \, e^x \cdot e^{yi} \qquad ( \text{for all }x, y \in \mathbb{R})\end{align*}

which is of course *consistent* with the original definition of $\displaystyle e^x$ on the **real numbers**, as it can be seen that for all $\displaystyle x \in \mathbb{R}$:

\begin{align*} e^{x} = e^{x+0i} = e^x \cdot e^{0i} = e^x \cdot e^0 = e^x \end{align*}

But then, how do we interpret this redefinition now that we are in the realm of complex numbers? For one, by **Euler’s Formula**, we know that $e^{yi}$ stands for the **unit complex number** with angle $y$. In addition, since multiplying a complex number by a *real positive constant* results in the rescaling of the number by that same constant, it becomes apparent that the natural exponential function — as defined above — maps a complex number $x+yi$ to another complex number whose **length** is $e^x$ and whose **angle** is $y$.

For example, $\displaystyle e^0=1, e^1=e, e^{\frac{\pi}{2}i}=i, e^{\pi i}=-1$ and $\displaystyle e^{1+ \pi i} = e^1 \cdot e^{\pi i} = -e$. In fact, one can also see that *by construction*, any **exponential** is necessarily a number with *non-zero* length (i.e., an exponential is always *non-zero*).

So the natural exponential function maps the entire complex plane to *non-zero* complex numbers, but perhaps what is more subtle is the fact that *any* non-zero complex number can be expressed as a power of $e$ as well. To see why, suppose that we are given a *non-zero* complex number $\displaystyle z$, then as long as we let:

- $x$ be the number such that $e^x$ is equal to the
**length**of $\displaystyle z$ (remember, this is always possible, since $z$ is a non-zero number, and hence must have non-zero length as well) - $y$ be
*an***angle**of $\displaystyle z$

then we will have that:

\begin{align*} e^{x+yi} & = e^x \cdot e^{yi} \\ & = \text{the complex number with the length and angle of }z \\ & = z \end{align*}

More specifically, given any *non-zero* complex number $\displaystyle z$, if $\displaystyle |z|$ stands for the **length** of $\displaystyle z$ and $\theta$ *an* **angle** of $\displaystyle z$, then we have just shown that:

\begin{align*} e^{\ln |z|+\theta i} & = e^{\ln |z|} \cdot e^{\theta i} \\ & = \text{the complex number with the length and angle of } z \\ & = z \end{align*}

In other words, we have just found *a* logarithm of $z$ — $\displaystyle \ln |z|+\theta i$ that is! However, notice the use of “*a*” instead of “*the*“, for as alluded to a bit earlier, this is not going to go all smoothly…

For one, since a complex number can have several **equivalent angles**, defining the natural exponential function on the *entire *complex plane can open up a *whole can of worms* — That is, a whole can of *infinitely-many* complex numbers whose exponentials are all but the same. For example:

\begin{align*} e^{\pi i}= e^{3 \pi i} = e^{5 \pi i} = e^{- \pi i} = \dots = -1 \end{align*}

which shows that the logarithm of $-1$ — or any other number for that matter — is ill-defined with this current setup.

Naturally, one way to remedy this *multivalue-ness* of logarithm is to *restrict* the domain of the exponential function to the **principal branch**. That is, the set of complex numbers whose **imaginary part** lies in the interval $\displaystyle (-\pi, \pi]$. With the domain restricted this way, we will then be able to prove that the exponentials of distinct numbers are *themselves* distinct, making the natural exponential an **invertible function **— mapping the principal branch to the set of *non-zero* complex numbers.

Under this setup, the inverse of the exponential function — the **natural logarithmic function **— maps the set of *non-zero* complex numbers to the principal branch. In fact, it can be shown that:

\boxed{\begin{equation*} \ln z = \displaystyle \ln |z| + (\arg{z}) i \ \end{equation*} }

where $\arg{z}$ stands for the **principal angle** of $\displaystyle z$ (i.e., the angle in the interval $\displaystyle (-\pi, \pi]$). For example:

\begin{align*} \ln (-1) = 0+\pi i \qquad \ln (-5) = \ln 5 + \pi i \qquad \ln (-1+i)= \ln (\sqrt{2}) + \frac{3\pi}{4}i \end{align*}

With the definition of natural logarithm now taken care of, we can proceed to define the **exponential function** of an *arbitrary base* $\displaystyle a$ — simply by *standardizing* it into base $\displaystyle e$:

\begin{align*} a^{z} = {\left(e^{\ln a}\right)}^{z} \, \stackrel{def}{=} \, e^{(\ln a)z} \end{align*}

Of course, in order for this definition to make sense, the base $\displaystyle a$ has to be *non-zero*. In addition, the base must not be $\displaystyle 1$ either, otherwise we will just end up with a **constant function** instead. Basically, this just means that the concept of **valid base** also has to be updated — from a *positive* *real number* that’s not $\displaystyle 1$, to a *non-zero complex number* that’s not $\displaystyle 1$. And that’s a great achievement if you think about it: the set of valid bases have just gone from the right side of a **real number line**, to almost the entire **complex plane**!

So all seems to be good. *Or is it?* Underneath the surface, the **properties of logarithm** are actually falling apart:

**Product Rule**fails: $\displaystyle \ln (-1 \cdot -1) = \ln 1 =0$, but $\displaystyle \ln (-1) + \ln (-1) = 2 \pi i$.**Reciprocal/Quotient Rule**fails: $\displaystyle \ln \left( \frac{1}{-1} \right) = \ln (-1) = \pi i$, but $\displaystyle – \ln(-1) = – \pi i$.**Power Rule**fails: $\displaystyle \ln \left[ (-1)^2 \right] = \ln 1 = 0$, but $\displaystyle 2 \ln (-1) = 2 \pi i$.**Root Rule**fails:**The Fundamental Theorem of Algebra**implies that a number can now have up to $n$ $n$^{th}complex roots, so the concept of*unique*$n$^{th}root is no longer applicable.

On a brighter side, **logarithmic functions** of an *arbitrary base* can now be defined — in terms of $\displaystyle \ln z$ — for all *non-zero* complex number, using an *instance* of what’s previously known as the **Change-of-Base Rule**:

\begin{align*} \log_a x \, \stackrel{def}{=} \, \frac{\ln x}{\ln a} \qquad (x \ne 0, a \text{ being a valid base}) \end{align*}

In light of this, it’s therefore no surprise that the full-fledged **Change-of-Base Rule** actually hold for complex logarithms in general:

\begin{align*} \log_a x & = \frac{\ln x}{\ln a} \\ & = \frac{\frac{\ln x}{\ln b}}{\frac{\ln a}{\ln b}} \\ & = \frac{\log_b x}{\log_b a} \qquad (x \ne 0, a \text{ and } b\text{ being both valid bases})\end{align*}

In particular, in the case where $x$ is also a *valid base*, we get that:

\begin{align*} \log_a x = \frac{\log_x x}{\log_x a} = \frac{1}{\log_x a} \end{align*}

What’s the name again? **Base-Swapping Rule** of course!

In addition, if we just start from the **Change-of-Base Rule** and solve for $\displaystyle \log_b x$, we get that the **Chain Rule** is here to stay as well:

\begin{align*} \log_b x = \log_b a \cdot \log_a x \qquad (x \ne 0, a \text{ and } b\text{ being both valid bases}) \end{align*}

In fact, it turns out that even the **Base-Argument Interchangeability** carries on as well, but only because of a *special instance* of “**Power Rule**” that we’ve built in into our definition of general exponential function:

\begin{align*} x^{\log_b y} & = e^{\ln x \cdot \log_b y} \\ & = e^{\frac{\ln x \cdot \ln y}{\ln b}} \\ & = e^{\ln y \cdot \log_b x} \\& =y^{\log_b x} \qquad (x, y \ne 0, b \text{ being a valid base}) \end{align*}

So that even though all properties of logarithm involving the *arguments* are lost in the process, those involving the *bases* all managed to come out of this unscathed.

What about the **properties of exponents** though? In two words: *not good*. 🙁

**Trivial Identities**holds: $\displaystyle a^0= 1, a^1=a$.**Additive Properties**holds: For all $\displaystyle z_1, z_2 \in \mathbb{C}$, $\displaystyle a^{z_1+ z_2} = a^{z_1} a^{z_2}$ and $\displaystyle a^{z_1 – z_2} = \frac{a^{z_1}}{a^{z_2}}$.**Common-Exponent Properties**fails: Given $a,b \ne 0$, we have that $\displaystyle (ab)^x \ne a^x b^x$ and $\displaystyle \left(\frac{a}{b}\right)^x \ne \frac{a^x}{b^x}$ in general.**Power Property**fails: $\displaystyle \left( a^{z_1}\right)^{z_2} = a^{z_1 \cdot z_2}$ when the*outer exponent*$z_2$ is an integer (thanks to the*Additive Properties*of exponent!), but is otherwise*false*in general.

Basically, every property — whose validity *depends* on the *argument-related* properties of logarithm — will fall out big time.

And finally, to ease the *information overload*, here’s a summary of the overall *gains* and *casualties* associated with expanding the domains of **exponential** and **logarithmic functions**:

- As long as we restrict to the domain of $\displaystyle e^z$ to a
**branch**, we can define logarithm on all*non-zero*complex numbers — and doing so using any*valid base*under the sun. - However, the five properties of logarithm involving the
*arguments*(e.g.,**Product Rule**,**Reciprocal Rule**,**Quotient Rule**,**Power Rule**,**Root Rule**) will be lost in the process. - Surprisingly though, the four properties of logarithm involving the
*bases*(i.e.,**Change-of-Base Rule**,**Chain Rule**,**Base-Swapping Rule**,**Base-Argument Interchangeability**) will be all preserved. - Due to the
*substantial loss*in the properties of logarithm, most of the**properties of exponent**will be falling apart as well.

So in retrospect, is it worth the effort going through the hurdle of expanding the domain of logarithmic functions? Heck, guess we’ll let you decide on that one! 🙂

Whew! Who would have thought that a pure venture into some basic theory can take us this far! Originally a tool for **computing large numbers** (e.g., turning products or quotients into sums or differences) and for **solving exponential equations** (through the **properties of logarithm**), logarithm has obviously gone a long way into finding itself in various branches of both *applied sciences* and *pure mathematics*.

For the applied scientists, logarithm tends to bring to mind topics such as **order-of-magnitude computation** (decimal or binary), **logarithmic scales** (e.g., **frequency in musical notes**, **Richter scale**, **pH**, **sound loudness**), **algorithmic complexities** and **log-normal distribution**. For others who live in the *ivory tower* (who’s that?), logarithm is often associated with *idealized objects* such as the **harmonic series**, **reciprocal funcion**, and even **prime numbers**,

In fact, when we toyed around the idea of incorporating** complex numbers** into logarithm, we found that not only were we able to extend the definition of logarithm to the **negative real numbers**, but to any other *non-zero* complex number as well. While the extension doesn’t always end up as the way we wanted, doing so actually paves the way of introducing complex logarithm into other *seemingly-unrelated* subjects, such as **integration by partial fractions in calculus** for instance.

And before someone’s brain *blows up*, here’s an **interactive table** for what we’ve found thus far:

Base |

Exponent |

Power |

(Order of) Magnitude |

Exponentiation |

Argument (of a Logarithm) |

Definition of Logarithm: $\displaystyle b^{\log_b x} = x \quad (x>0)$ |

Inverse Relationship Between Exponential and Logarithmic Function: $\displaystyle \log_b (b^x) = x$ |

Trivial Logarithmic Identities: $\displaystyle \log_b 1 = 0$, $\displaystyle \log_b b = 1$, $\displaystyle \log_b \frac{1}{b} = -1$ |

$\displaystyle \log x$ |

Number of Digits (Decimal Representation) |

Characteristic |

Mantissa |

$\displaystyle \log_2 x$ |

Number of Bits (Binary Representation) |

Binary Tree |

Number of Required Operations (Algorithm) |

Octave (Music) |

Frequency (Musical Notes) |

$\displaystyle \ln x$ |

Reciprocal Function |

Natural Exponential Function |

Harmonic Series |

$\displaystyle \log x$ (Notational Ambiguity) |

Valid Base |

Large Base |

Small Base |

Graph (Logarithmic Functions) |

Linear Scale vs. Logarithmic Scale |

Exponential Growth/Shrinking |

Frequency of Musical Notes (Music Theory): Relative note frequencies converted into binary logarithm. Increase a semitone increases the frequency of a note by $\displaystyle \sqrt[12]{2}$. |

pH (Chemistry): Concentration of hydrogen ions converted into negative (common) logarithm. Increase pH by $1$ decreases the acidity of a solution by tenfold. |

Richter Scale (Seismology): Relative seismic wave amplitude converted into common logarithm. One unit of increase in Richter scale increases the severity of an earthquake by tenfold. |

Decibel (Acoustics): Relative sound power converted into common logarithm (with the unit in bel), where one bel of increase boosts the relative sound power by tenfold. |

Assuming that all logarithms below are defined under a *valid* base $b$, then the following properties hold:

Product Rule: $\displaystyle \log (xy) =\log x + \log y \quad (x,y >0)$ |

Reciprocal Rule: $\displaystyle \log \left(\frac{1}{x} \right) = – \log x \quad (x>0)$ |

Quotient Rule: $\displaystyle \log \left( \frac{x}{y} \right) =\log x – \log y \quad (x,y>0)$ |

Power Rule: $\displaystyle \log (x^n) = n \log x \quad (x>0, n \in \mathbb{Z})$ |

Root Rule: $\displaystyle \log (\sqrt[n]{x}) = \frac{\log x}{n} \quad (x>0, n \in \mathbb{N})$ |

Generalized Power Rule: $\displaystyle \log (x^p) = p \log x \quad (x>0, p \in \mathbb{R})$ |

Given two *valid* bases $a, b$, and that all logarithms with no *explicit base* are defined under the base $b$, then the following properties hold:

Chain Rule: $\displaystyle \log x = \log a \cdot \log_a x \quad (x>0)$ |

Change-of-Base Rule: $\displaystyle \log_a x = \frac{\log x}{\log a} \quad (x>0)$ |

Base-Swapping Rule: $\displaystyle \log_a x = \frac{1}{\log_x a} \quad (x \text{ being also a valid base})$ |

Base-Argument Interchangeability: $\displaystyle x^{\log y} = y^{\log x} \quad (x,y >0)$ |

Natural Exponential Function: $\displaystyle e^{x+yi} \, \stackrel{def}{=} \, e^x \cdot e^{yi} \quad ( \text{for all }x, y \in \mathbb{R})$ |

Natural Logarithmic Function: For any non-zero complex number $\displaystyle z$, $\ln z \, \stackrel{def}{=} \, \ln |z| + (\arg{z}) i$, where $\displaystyle |z|$ stands for the length of $\displaystyle z$, and $\displaystyle \arg{z}$ the principal angle of $\displaystyle z$ in the interval $\displaystyle (-\pi, \pi]$. |

Valid Base: Changes from a positive real number that is not equal to one, to any non-zero complex number that is not equal to one. |

Exponential Function (Arbitrary Base): $\displaystyle a^{z} \, \stackrel{def}{=} \, e^{(\ln a)z} \quad (a \text{ being a valid base})$ |

Logarithmic Function (Arbitrary Base): $\displaystyle \log_a x \, \stackrel{def}{=} \, \frac{\ln x}{\ln a} \qquad (x \ne 0, a \text{ being a valid base})$ |

Properties of Logarithm: All rules involving the arguments fall apart (i.e., Product Rule, Reciprocal Rule, Quotient Rule, Power Rule and Root Rule). On the other hand, all rules involving the bases are preserved (i.e., Chain Rule, Change-of-Base Rule, Base-Swapping Rule, Base-Argument Interchangeability) |

Properties of Exponents: Trivial Properties (i.e. $\displaystyle a^0=1, a^1=a$) and Additive Properties (i.e., $\displaystyle a^{z_1+ z_2} = a^{z_1} a^{z_2}, \, a^{z_1 – z_2} = \frac{a^{z_1}}{a^{z_2}}$) remain. All other properties which rely on the argument-related properties of logarithm fail (e.g., Common-Exponent Properties, Power Property). |

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