Precalculus Examples

$\log_{b} (\frac{\sqrt{x y^{4}}}{z^{4}})$

Step 1

Rewrite $\log_{b} (\frac{\sqrt{x y^{4}}}{z^{4}})$ as $\log_{b} (\sqrt{x y^{4}}) - \log_{b} (z^{4})$ .

$\log_{b} (\sqrt{x y^{4}}) - \log_{b} (z^{4})$

Step 2

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

$\log_{b} ({(x y^{4})}^{\frac{1}{2}}) - \log_{b} (z^{4})$

Step 3

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

$\frac{1}{2} \log_{b} (x y^{4}) - \log_{b} (z^{4})$

Step 4

Expand $\log_{b} (z^{4})$ by moving $4$ outside the logarithm.

$\frac{1}{2} \log_{b} (x y^{4}) - (4 \log_{b} (z))$

Step 5

Multiply $4$ by $- 1$ .

$\frac{1}{2} \log_{b} (x y^{4}) - 4 \log_{b} (z)$

Step 6

Rewrite $\log_{b} (x y^{4})$ as $\log_{b} (x) + \log_{b} (y^{4})$ .

$\frac{1}{2} (\log_{b} (x) + \log_{b} (y^{4})) - 4 \log_{b} (z)$

Step 7

Expand $\log_{b} (y^{4})$ by moving $4$ outside the logarithm.

$\frac{1}{2} (\log_{b} (x) + 4 \log_{b} (y)) - 4 \log_{b} (z)$

Step 8

Simplify each term.

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Step 8.1

Apply the distributive property.

$\frac{1}{2} \log_{b} (x) + \frac{1}{2} (4 \log_{b} (y)) - 4 \log_{b} (z)$

Step 8.2

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

$\frac{\log_{b} (x)}{2} + \frac{1}{2} (4 \log_{b} (y)) - 4 \log_{b} (z)$

Step 8.3

Cancel the common factor of $2$ .

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Step 8.3.1

Factor $2$ out of $4 \log_{b} (y)$ .

$\frac{\log_{b} (x)}{2} + \frac{1}{2} (2 (2 \log_{b} (y))) - 4 \log_{b} (z)$

Step 8.3.2

Cancel the common factor.

$\frac{\log_{b} (x)}{2} + \frac{1}{2} (2 (2 \log_{b} (y))) - 4 \log_{b} (z)$

Step 8.3.3

Rewrite the expression.

$\frac{\log_{b} (x)}{2} + 2 \log_{b} (y) - 4 \log_{b} (z)$