# how to solve inequality using logarithm

I was given the following expression :$$(0.87)^n\leq 0.1$$

And the next step was: $$n\geq \frac{log(0.1)}{log(0.87)}$$

What was the steps betweens?

• $log(0,87) <0$ ,multiplying by a (-) turn the sign of inequality – Khosrotash Jul 9 '15 at 13:11

## 4 Answers

They took the $log$ of both sides:

$log((0.87)^n) \leq log(0.1)$.

Note then that there is then the usage of the rule:

$log(a^b) = b \cdot log(a)$,

and further note the sign of $log(0.87)$ when dividing the equation through by it.

$$log(0.87)^n\leq log (0.1) ,$$ and then $$n \cdot log(0.87)\leq log (0.1) .$$ Take it from here.

If you have $$0.87^n \leq 1$$ then you can take the logarithm of both sides, since the logarithm is something we call strictly increasing. That just means that if you have $a < b$, then $\ln a < \ln b$ is true as well. In any case, you get $$\log 0.87^n \leq \log 0.1$$ The power rule says you can pull the $n$ infront to get $$n \log 0.87 \leq \log 0.1$$

Now you divide through by $\ln 0.87$, which is negative, so you flip the inequality sign to get $$n \geq \frac{\log 0.1}{\log 0.87}$$

In general, $\ln a$ is negative if $a < 1$.

Take logs on both sides, you have $(0.87)^n \leq 0.1\implies n\cdot log(0.87) \leq log(0.1)\implies n\geq \frac{log(0.1)}{log(0.87)}$ Note that the direction f the inequality changes because log(0.87) is negative .