Precalculus Examples

$\log_{3} (5) \cdot \log_{25} (9)$

Step 1

Rewrite $\log_{3} (5)$ using the change of base formula.

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Step 1.1

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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{25} (9)$

Step 1.2

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

$\frac{\log (5)}{\log (3)} \log_{25} (9)$

Step 2

Rewrite $\log_{25} (9)$ using the change of base formula.

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Step 2.1

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 $0$ .

$\frac{\log (5)}{\log (3)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 2.2

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

$\frac{\log (5)}{\log (3)} \cdot \frac{\log (9)}{\log (25)}$

Step 3

Rewrite $\log (9)$ as $\log (3^{2})$ .

$\frac{\log (5)}{\log (3)} \cdot \frac{\log (3^{2})}{\log (25)}$

Step 4

Expand $\log (3^{2})$ by moving $2$ outside the logarithm.

$\frac{\log (5)}{\log (3)} \cdot \frac{2 \log (3)}{\log (25)}$

Step 5

Rewrite $\log (25)$ as $\log (5^{2})$ .

$\frac{\log (5)}{\log (3)} \cdot \frac{2 \log (3)}{\log (5^{2})}$

Step 6

Expand $\log (5^{2})$ by moving $2$ outside the logarithm.

$\frac{\log (5)}{\log (3)} \cdot \frac{2 \log (3)}{2 \log (5)}$

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Cancel the common factor.

$\frac{\log (5)}{\log (3)} \cdot \frac{2 \log (3)}{2 \log (5)}$

Step 7.2

Rewrite the expression.

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

Step 8

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

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

$\frac{1}{\log (3)} \log (3)$

Step 9

Cancel the common factor of $\log (3)$ .

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Step 9.1

Cancel the common factor.

$\frac{1}{\log (3)} \log (3)$

Step 9.2

Rewrite the expression.

$1$