Precalculus Examples

$\log (\sqrt{\frac{x y^{9}}{y^{4}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x y^{9}}{y^{4}}}$ as ${(\frac{x y^{9}}{y^{4}})}^{\frac{1}{2}}$ .

$\log ({(\frac{x y^{9}}{y^{4}})}^{\frac{1}{2}})$

Step 2

Expand $\log ({(\frac{x y^{9}}{y^{4}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (\frac{x y^{9}}{y^{4}})$

Step 3

Reduce the expression $\frac{x y^{9}}{y^{4}}$ by cancelling the common factors.

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Step 3.1

Factor $y^{4}$ out of $x y^{9}$ .

$\frac{1}{2} \log (\frac{y^{4} (x y^{5})}{y^{4}})$

Step 3.2

Multiply by $1$ .

$\frac{1}{2} \log (\frac{y^{4} (x y^{5})}{y^{4} \cdot 1})$

Step 3.3

Cancel the common factor.

$\frac{1}{2} \log (\frac{y^{4} (x y^{5})}{y^{4} \cdot 1})$

Step 3.4

Rewrite the expression.

$\frac{1}{2} \log (\frac{x y^{5}}{1})$

Step 4

Divide $x y^{5}$ by $1$ .

$\frac{1}{2} \log (x y^{5})$

Step 5

Rewrite $\log (x y^{5})$ as $\log (x) + \log (y^{5})$ .

$\frac{1}{2} (\log (x) + \log (y^{5}))$

Step 6

Expand $\log (y^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{2} (\log (x) + 5 \log (y))$

Step 7

Apply the distributive property.

$\frac{1}{2} \log (x) + \frac{1}{2} (5 \log (y))$

Step 8

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

$\frac{\log (x)}{2} + \frac{1}{2} (5 \log (y))$

Step 9

Multiply $\frac{1}{2} (5 \log (y))$ .

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Step 9.1

Combine $5$ and $\frac{1}{2}$ .

$\frac{\log (x)}{2} + \frac{5}{2} \log (y)$

Step 9.2

Combine $\frac{5}{2}$ and $\log (y)$ .

$\frac{\log (x)}{2} + \frac{5 \log (y)}{2}$