Evaluate log base 0.2 of 125

$\log_{0.2} (125)$

Step 1

Rewrite $\log_{0.2} (125)$ using the change of base formula.

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (125)}{\log (0.2)}$

Step 2

Log base $10$ of $0.2$ is approximately $- 0.69897$ .

$\frac{\log (125)}{- 0.69897}$

Step 3

Move the negative in front of the fraction.

$- \frac{\log (125)}{0.69897}$

Step 4

Log base $10$ of $125$ is approximately $2.09691001$ .

$- \frac{2.09691001}{0.69897}$

Step 5

Divide $2.09691001$ by $0.69897$ .

$- 1 \cdot 3$

Step 6

Multiply $- 1$ by $3$ .

$- 3$