Precalculus Examples

$(\log_{9} (7)) (\log_{7} (10))$

Step 1

Multiply $\log_{9} (7)$ by $\log_{7} (10)$ .

$\log_{9} (7) \log_{7} (10)$

Step 2

Rewrite $\log_{9} (7)$ 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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{7} (10)$

Step 2.2

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

$\frac{\log (7)}{\log (9)} \log_{7} (10)$

Step 3

Rewrite $\log_{7} (10)$ 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 $0$ .

$\frac{\log (7)}{\log (9)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 3.2

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

$\frac{\log (7)}{\log (9)} \cdot \frac{\log (10)}{\log (7)}$

Step 4

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

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Step 4.1

Logarithm base $10$ of $10$ is $1$ .

$\frac{\log (7)}{\log (9)} \cdot \frac{1}{\log (7)}$

Step 4.2

Cancel the common factor.

$\frac{\log (7)}{\log (9)} \cdot \frac{1}{\log (7)}$

Step 4.3

Rewrite the expression.

$\frac{1}{\log (9)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{\log (9)}$

Decimal Form:

$1.04795163 \dots$