Precalculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

Apply the product rule to $a^{2} \sqrt{b}$ .

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

Step 1.2

Multiply the exponents in ${(a^{2})}^{4}$ .

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Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.2

Multiply $2$ by $4$ .

$\log_{2} (a^{8} {\sqrt{b}}^{4})$

Step 2

Rewrite ${\sqrt{b}}^{4}$ as $b^{2}$ .

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Step 2.1

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

$\log_{2} (a^{8} {(b^{\frac{1}{2}})}^{4})$

Step 2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{2} (a^{8} b^{\frac{1}{2} \cdot 4})$

Step 2.3

Combine $\frac{1}{2}$ and $4$ .

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

Step 2.4

Cancel the common factor of $4$ and $2$ .

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Step 2.4.1

Factor $2$ out of $4$ .

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

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $2$ out of $2$ .

$\log_{2} (a^{8} b^{\frac{2 \cdot 2}{2 (1)}})$

Step 2.4.2.2

Cancel the common factor.

$\log_{2} (a^{8} b^{\frac{2 \cdot 2}{2 \cdot 1}})$

Step 2.4.2.3

Rewrite the expression.

$\log_{2} (a^{8} b^{\frac{2}{1}})$

Step 2.4.2.4

Divide $2$ by $1$ .

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

Step 3

Rewrite $\log_{2} (a^{8} b^{2})$ 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 $0$ .

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

Step 3.2

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

$\frac{\log (a^{8} b^{2})}{\log (2)}$