Precalculus Examples

$\log (\sqrt{a^{2} b^{3}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a^{2} b^{3}}$ as ${(a^{2} b^{3})}^{\frac{1}{2}}$ .

$\log ({(a^{2} b^{3})}^{\frac{1}{2}})$

Step 2

Expand $\log ({(a^{2} b^{3})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (a^{2} b^{3})$

Step 3

Rewrite $\log (a^{2} b^{3})$ as $\log (a^{2}) + \log (b^{3})$ .

$\frac{1}{2} (\log (a^{2}) + \log (b^{3}))$

Step 4

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

$\frac{1}{2} (2 \log (a) + \log (b^{3}))$

Step 5

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

$\frac{1}{2} (2 \log (a) + 3 \log (b))$

Step 6

Apply the distributive property.

$\frac{1}{2} (2 \log (a)) + \frac{1}{2} (3 \log (b))$

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Factor $2$ out of $2 \log (a)$ .

$\frac{1}{2} (2 (\log (a))) + \frac{1}{2} (3 \log (b))$

Step 7.2

Cancel the common factor.

$\frac{1}{2} (2 \log (a)) + \frac{1}{2} (3 \log (b))$

Step 7.3

Rewrite the expression.

$\log (a) + \frac{1}{2} (3 \log (b))$

Step 8

Multiply $\frac{1}{2} (3 \log (b))$ .

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Step 8.1

Combine $3$ and $\frac{1}{2}$ .

$\log (a) + \frac{3}{2} \log (b)$

Step 8.2

Combine $\frac{3}{2}$ and $\log (b)$ .

$\log (a) + \frac{3 \log (b)}{2}$