Precalculus Examples

$(\log_{9} (x)) \cdot (\log_{x} (9))$

Step 1

Multiply $\log_{9} (x)$ by $\log_{x} (9)$ .

$\log_{9} (x) (\log_{x} (9))$

Step 2

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

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

Step 2.2

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

$\frac{\log (x)}{\log (9)} \log_{x} (9)$

Step 3

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

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

Step 3.2

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

$\frac{\log (x)}{\log (9)} \cdot \frac{\log (9)}{\log (x)}$

Step 4

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

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Step 4.1

Cancel the common factor.

$\frac{\log (x)}{\log (9)} \cdot \frac{\log (9)}{\log (x)}$

Step 4.2

Rewrite the expression.

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

Step 5

Cancel the common factor of $\log (9)$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$1$