Precalculus Examples

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

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{b}$ as $b^{\frac{1}{2}}$ .

$\log_{2} ({(a^{2} b^{\frac{1}{2}})}^{4})$

Step 2

Expand $\log_{2} ({(a^{2} b^{\frac{1}{2}})}^{4})$ by moving $4$ outside the logarithm.

$4 \log_{2} (a^{2} b^{\frac{1}{2}})$

Step 3

Rewrite $\log_{2} (a^{2} b^{\frac{1}{2}})$ as $\log_{2} (a^{2}) + \log_{2} (b^{\frac{1}{2}})$ .

$4 (\log_{2} (a^{2}) + \log_{2} (b^{\frac{1}{2}}))$

Step 4

Expand $\log_{2} (a^{2})$ by moving $2$ outside the logarithm.

$4 (2 \log_{2} (a) + \log_{2} (b^{\frac{1}{2}}))$

Step 5

Expand $\log_{2} (b^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$4 (2 \log_{2} (a) + \frac{1}{2} \log_{2} (b))$

Step 6

Combine $\frac{1}{2}$ and $\log_{2} (b)$ .

$4 (2 \log_{2} (a) + \frac{\log_{2} (b)}{2})$

Step 7

Apply the distributive property.

$4 (2 \log_{2} (a)) + 4 \frac{\log_{2} (b)}{2}$

Step 8

Multiply $2$ by $4$ .

$8 \log_{2} (a) + 4 \frac{\log_{2} (b)}{2}$

Step 9

Cancel the common factor of $2$ .

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Step 9.1

Factor $2$ out of $4$ .

$8 \log_{2} (a) + 2 (2) \frac{\log_{2} (b)}{2}$

Step 9.2

Cancel the common factor.

$8 \log_{2} (a) + 2 \cdot 2 \frac{\log_{2} (b)}{2}$

Step 9.3

Rewrite the expression.

$8 \log_{2} (a) + 2 \log_{2} (b)$