Precalculus Examples

$(\log_{7} (2)) (\log_{2} (49))$

Step 1

Multiply $\log_{7} (2)$ by $\log_{2} (49)$ .

$\log_{7} (2) \log_{2} (49)$

Step 2

Rewrite $\log_{7} (2)$ 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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{2} (49)$

Step 2.2

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

$\frac{\log (2)}{\log (7)} \log_{2} (49)$

Step 3

Rewrite $\log_{2} (49)$ 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 $0$ .

$\frac{\log (2)}{\log (7)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 3.2

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

$\frac{\log (2)}{\log (7)} \cdot \frac{\log (49)}{\log (2)}$

Step 4

Cancel the common factor of $\log (2)$ .

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Step 4.1

Cancel the common factor.

$\frac{\log (2)}{\log (7)} \cdot \frac{\log (49)}{\log (2)}$

Step 4.2

Rewrite the expression.

$\frac{1}{\log (7)} \log (49)$

Step 5

Combine $\frac{1}{\log (7)}$ and $\log (49)$ .

$\frac{\log (49)}{\log (7)}$

Step 6

Rewrite $\log (49)$ as $\log (7^{2})$ .

$\frac{\log (7^{2})}{\log (7)}$

Step 7

Expand $\log (7^{2})$ by moving $2$ outside the logarithm.

$\frac{2 \log (7)}{\log (7)}$

Step 8

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

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Step 8.1

Cancel the common factor.

$\frac{2 \log (7)}{\log (7)}$

Step 8.2

Divide $2$ by $1$ .

$2$