Precalculus Examples

$\log_{2} (6) \cdot \log_{6} (8)$

Step 1

Rewrite $\log_{2} (6)$ 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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{6} (8)$

Step 1.2

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

$\frac{\log (6)}{\log (2)} \log_{6} (8)$

Step 2

Rewrite $\log_{6} (8)$ 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 $0$ .

$\frac{\log (6)}{\log (2)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 2.2

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

$\frac{\log (6)}{\log (2)} \cdot \frac{\log (8)}{\log (6)}$

Step 3

Cancel the common factor of $\log (6)$ .

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Step 3.1

Cancel the common factor.

$\frac{\log (6)}{\log (2)} \cdot \frac{\log (8)}{\log (6)}$

Step 3.2

Rewrite the expression.

$\frac{1}{\log (2)} \log (8)$

Step 4

Combine $\frac{1}{\log (2)}$ and $\log (8)$ .

$\frac{\log (8)}{\log (2)}$

Step 5

Rewrite $\log (8)$ as $\log (2^{3})$ .

$\frac{\log (2^{3})}{\log (2)}$

Step 6

Expand $\log (2^{3})$ by moving $3$ outside the logarithm.

$\frac{3 \log (2)}{\log (2)}$

Step 7

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

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Step 7.1

Cancel the common factor.

$\frac{3 \log (2)}{\log (2)}$

Step 7.2

Divide $3$ by $1$ .

$3$