Precalculus Examples

$\log_{h} (\frac{\sqrt{x y^{3}}}{z^{3}})$

Step 1

Rewrite $\log_{h} (\frac{\sqrt{x y^{3}}}{z^{3}})$ as $\log_{h} (\sqrt{x y^{3}}) - \log_{h} (z^{3})$ .

$\log_{h} (\sqrt{x y^{3}}) - \log_{h} (z^{3})$

Step 2

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

$\log_{h} ({(x y^{3})}^{\frac{1}{2}}) - \log_{h} (z^{3})$

Step 3

Expand $\log_{h} ({(x y^{3})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{h} (x y^{3}) - \log_{h} (z^{3})$

Step 4

Expand $\log_{h} (z^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{2} \log_{h} (x y^{3}) - (3 \log_{h} (z))$

Step 5

Multiply $3$ by $- 1$ .

$\frac{1}{2} \log_{h} (x y^{3}) - 3 \log_{h} (z)$

Step 6

Rewrite $\log_{h} (x y^{3})$ as $\log_{h} (x) + \log_{h} (y^{3})$ .

$\frac{1}{2} (\log_{h} (x) + \log_{h} (y^{3})) - 3 \log_{h} (z)$

Step 7

Expand $\log_{h} (y^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{2} (\log_{h} (x) + 3 \log_{h} (y)) - 3 \log_{h} (z)$

Step 8

Simplify each term.

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Step 8.1

Apply the distributive property.

$\frac{1}{2} \log_{h} (x) + \frac{1}{2} (3 \log_{h} (y)) - 3 \log_{h} (z)$

Step 8.2

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

$\frac{\log_{h} (x)}{2} + \frac{1}{2} (3 \log_{h} (y)) - 3 \log_{h} (z)$

Step 8.3

Multiply $\frac{1}{2} (3 \log_{h} (y))$ .

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Step 8.3.1

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

$\frac{\log_{h} (x)}{2} + \frac{3}{2} \log_{h} (y) - 3 \log_{h} (z)$

Step 8.3.2

Combine $\frac{3}{2}$ and $\log_{h} (y)$ .

$\frac{\log_{h} (x)}{2} + \frac{3 \log_{h} (y)}{2} - 3 \log_{h} (z)$