Precalculus Examples

$\log_{5} (22) \cdot \log_{22} (125)$

Step 1

Rewrite $\log_{5} (22)$ using the change of base formula.

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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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{22} (125)$

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

$\frac{\log (22)}{\log (5)} \log_{22} (125)$

Step 2

Rewrite $\log_{22} (125)$ using the change of base formula.

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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 $0$ .

$\frac{\log (22)}{\log (5)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

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

$\frac{\log (22)}{\log (5)} \cdot \frac{\log (125)}{\log (22)}$

Step 3

Cancel the common factor of $\log (22)$ .

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Cancel the common factor.

$\frac{\log (22)}{\log (5)} \cdot \frac{\log (125)}{\log (22)}$

Rewrite the expression.

$\frac{1}{\log (5)} \log (125)$

Step 4

Combine $\frac{1}{\log (5)}$ and $\log (125)$ .

$\frac{\log (125)}{\log (5)}$

Step 5

Rewrite $\log (125)$ as $\log (5^{3})$ .

$\frac{\log (5^{3})}{\log (5)}$

Step 6

Expand $\log (5^{3})$ by moving $3$ outside the logarithm.

$\frac{3 \log (5)}{\log (5)}$

Step 7

Cancel the common factor of $\log (5)$ .

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Cancel the common factor.

$\frac{3 \log (5)}{\log (5)}$

Divide $3$ by $1$ .

$3$