Precalculus Examples

$3 \log_{0.2} (5)$

Step 1

Simplify $3 \log_{0.2} (5)$ by moving $3$ inside the logarithm.

$\log_{0.2} (5^{3})$

Step 2

Raise $5$ to the power of $3$ .

$\log_{0.2} (125)$

Step 3

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

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Step 3.1

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 $0$ .

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

Step 3.2

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

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

Step 4

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

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

Step 5

Move the negative in front of the fraction.

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

Step 6

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

$- \frac{2.09691001}{0.69897}$

Step 7

Divide $2.09691001$ by $0.69897$ .

$- 1 \cdot 3$

Step 8

Multiply $- 1$ by $3$ .

$- 3$