Precalculus Examples

$\log_{7} (3) \log_{3} (8)$

Step 1

Rewrite $\log_{7} (3)$ 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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{3} (8)$

Step 1.2

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

$\frac{\log (3)}{\log (7)} \log_{3} (8)$

Step 2

Rewrite $\log_{3} (8)$ 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 $0$ .

$\frac{\log (3)}{\log (7)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 2.2

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

$\frac{\log (3)}{\log (7)} \cdot \frac{\log (8)}{\log (3)}$

Step 3

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

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Step 3.1

Cancel the common factor.

$\frac{\log (3)}{\log (7)} \cdot \frac{\log (8)}{\log (3)}$

Step 3.2

Rewrite the expression.

$\frac{1}{\log (7)} \log (8)$

Step 4

Combine $\frac{1}{\log (7)}$ and $\log (8)$ .

$\frac{\log (8)}{\log (7)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (8)}{\log (7)}$

Decimal Form:

$1.06862156 \dots$