Precalculus Examples

$\log_{4} (7) \cdot \log_{7} (16)$

Step 1

Rewrite $\log_{4} (7)$ 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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{7} (16)$

Step 1.2

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

$\frac{\log (7)}{\log (4)} \log_{7} (16)$

Step 2

Rewrite $\log_{7} (16)$ 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 $0$ .

$\frac{\log (7)}{\log (4)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 2.2

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

$\frac{\log (7)}{\log (4)} \cdot \frac{\log (16)}{\log (7)}$

Step 3

Cancel the common factor of $\log (7)$ .

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Step 3.1

Cancel the common factor.

$\frac{\log (7)}{\log (4)} \cdot \frac{\log (16)}{\log (7)}$

Step 3.2

Rewrite the expression.

$\frac{1}{\log (4)} \log (16)$

Step 4

Combine $\frac{1}{\log (4)}$ and $\log (16)$ .

$\frac{\log (16)}{\log (4)}$

Step 5

Rewrite $\log (16)$ as $\log (2^{4})$ .

$\frac{\log (2^{4})}{\log (4)}$

Step 6

Expand $\log (2^{4})$ by moving $4$ outside the logarithm.

$\frac{4 \log (2)}{\log (4)}$

Step 7

Rewrite $\log (4)$ as $\log (2^{2})$ .

$\frac{4 \log (2)}{\log (2^{2})}$

Step 8

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

$\frac{4 \log (2)}{2 \log (2)}$

Step 9

Cancel the common factor of $4$ and $2$ .

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Step 9.1

Factor $2$ out of $4 \log (2)$ .

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

Step 9.2

Cancel the common factor.

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

Step 9.3

Rewrite the expression.

$2$