Precalculus Examples

$\frac{\log_{b} (a) \cdot \log_{a} (b)}{\log (10)}$

Step 1

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

$\frac{\log_{b} (a) \cdot \log_{a} (b)}{1}$

Step 2

Divide $\log_{b} (a) \cdot \log_{a} (b)$ by $1$ .

$\log_{b} (a) \log_{a} (b)$

Step 3

Rewrite $\log_{b} (a)$ 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 $0$ .

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

Step 3.2

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

$\frac{\log (a)}{\log (b)} \log_{a} (b)$

Step 4

Rewrite $\log_{a} (b)$ using the change of base formula.

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Step 4.1

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 $0$ .

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

Step 4.2

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

$\frac{\log (a)}{\log (b)} \cdot \frac{\log (b)}{\log (a)}$

Step 5

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

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Step 5.1

Cancel the common factor.

$\frac{\log (a)}{\log (b)} \cdot \frac{\log (b)}{\log (a)}$

Step 5.2

Rewrite the expression.

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$1$