Precalculus Examples

$(\log_{3} (25)) (\log_{5} (27))$

Step 1

Multiply $\log_{3} (25)$ by $\log_{5} (27)$ .

$\log_{3} (25) \log_{5} (27)$

Step 2

Rewrite $\log_{3} (25)$ 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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{5} (27)$

Step 2.2

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

$\frac{\log (25)}{\log (3)} \log_{5} (27)$

Step 3

Rewrite $\log_{5} (27)$ 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 $0$ .

$\frac{\log (25)}{\log (3)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 3.2

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

$\frac{\log (25)}{\log (3)} \cdot \frac{\log (27)}{\log (5)}$

Step 4

Multiply $\frac{\log (25)}{\log (3)}$ by $\frac{\log (27)}{\log (5)}$ .

$\frac{\log (25) \log (27)}{\log (3) \log (5)}$

Step 5

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

$\frac{\log (5^{2}) \log (27)}{\log (3) \log (5)}$

Step 6

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

$\frac{2 \log (5) \log (27)}{\log (3) \log (5)}$

Step 7

Rewrite $\log (27)$ as $\log (3^{3})$ .

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

Step 8

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

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

Step 9

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

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Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

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

Step 10

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

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Step 10.1

Cancel the common factor.

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

Step 10.2

Divide $2 \cdot (3)$ by $1$ .

$2 \cdot (3)$

Step 11

Multiply $2$ by $3$ .

$6$