Precalculus Examples

$\log_{b} (\sqrt{b^{9}})$

Step 1

Rewrite $b^{9}$ as ${(b^{4})}^{2} b$ .

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Step 1.1

Factor out $b^{8}$ .

$\log_{b} (\sqrt{b^{8} b})$

Step 1.2

Rewrite $b^{8}$ as ${(b^{4})}^{2}$ .

$\log_{b} (\sqrt{{(b^{4})}^{2} b})$

Step 2

Pull terms out from under the radical.

$\log_{b} (b^{4} \sqrt{b})$

Step 3

Rewrite $\log_{b} (b^{4} \sqrt{b})$ as $\log_{b} (b^{4}) + \log_{b} (\sqrt{b})$ .

$\log_{b} (b^{4}) + \log_{b} (\sqrt{b})$

Step 4

Use logarithm rules to move $4$ out of the exponent.

$4 \log_{b} (b) + \log_{b} (\sqrt{b})$

Step 5

Logarithm base $b$ of $b$ is $1$ .

$4 \cdot 1 + \log_{b} (\sqrt{b})$

Step 6

Multiply $4$ by $1$ .

$4 + \log_{b} (\sqrt{b})$