Precalculus Examples

$\log_{5} (\sqrt{x^{5} y})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{5} y}$ as ${(x^{5} y)}^{\frac{1}{2}}$ .

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

Step 2

Expand $\log_{5} ({(x^{5} y)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 3

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

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

Step 4

Expand $\log_{5} (x^{5})$ by moving $5$ outside the logarithm.

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

Step 5

Apply the distributive property.

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

Step 6

Multiply $\frac{1}{2} (5 \log_{5} (x))$ .

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Step 6.1

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

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

Step 6.2

Combine $\frac{5}{2}$ and $\log_{5} (x)$ .

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

Step 7

Combine $\frac{1}{2}$ and $\log_{5} (y)$ .

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