Logarithmic Inequality --- How do I solve this: Log base 0. 3 (x^2+8) > log base 0. 3 (9x)? | Math Vault
Menu

# Logarithmic Inequality — How do I solve this: Log base 0. 3 (x^2+8) > log base 0. 3 (9x)?

Subham asked the following question involving a logarithmic inequality:

Solve $\log_{0.3} (x^2+8) > \log_{0.3} (9x)$.

And here is our answer:

## A Brush-Up on Small-Base Logarithmic Function

When $\log(x)$ has a base that is strictly between $0$ and $1$, it behaves as a strictly decreasing function, meaning that:

$\displaystyle x_1 > x_2 \iff \log x_1 < \log x_2$

## Proof

### Preliminary fact about Small-Base Exponential Function

First, notice that while for large bases ($>1$) exponential functions are strictly increasing, for small bases (i.e., base strictly between $0$ and $1$), exponential functions are actually strictly decreasing. That is:

For all $x_1, x_2 \in \mathbb{R}$ and any small base $b$, $x_1 > x_2 \iff b^{x_1} < b^{x_2}$.

### Going Back to the Proof

Given any two positive real numbers $x_1, x_2$ and any small base $b$, we break down the proof in two parts:

1. $\log_b (x_1) < \log_b (x_2) \longrightarrow x_1 > x_2$
2. $\log_b (x_1) \ge \log_b (x_2) \longrightarrow x_1 \le x_2$

For the first part, suppose that $\log (x_1) < \log (x_2)$, then the strictly-decreasing-ness of the exponential function with base $b$ implies that:

$\displaystyle b^{\log_b (x_1)} > b^{\log_b (x_2)}$

Or equivalently,

$\displaystyle x_1 > x_2$

So the first part is done. The second part is proved similarly.  $\blacksquare$

## Going Back to the Original Logarithmic Inequality

Note that the logarithmic function of base 0.3 is strictly decreasing.

Applying the strictly-decreasing-ness of $\log_{0.3} (x)$ to the original logarithmic inequality, we have:

$x^2 + 8 < 9$

or

$(x-1)(x-8) < 0$

which means that $1 < x < 8$. Pretty neat. Isn’t it? 🙂

Follow

#### About the Author

Math Vault and its Redditbots has the singular goal of advocating for education in higher mathematics through digital publishing and the uncanny use of technologies. Head to the Vault for more math cookies. :)