Precalculus Examples

$\log_{b} (\sqrt{\frac{x^{3} y^{2}}{z^{4}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x^{3} y^{2}}{z^{4}}}$ as ${(\frac{x^{3} y^{2}}{z^{4}})}^{\frac{1}{2}}$ .

$\log_{b} ({(\frac{x^{3} y^{2}}{z^{4}})}^{\frac{1}{2}})$

Step 2

Expand $\log_{b} ({(\frac{x^{3} y^{2}}{z^{4}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{b} (\frac{x^{3} y^{2}}{z^{4}})$

Step 3

Rewrite $\log_{b} (\frac{x^{3} y^{2}}{z^{4}})$ as $\log_{b} (x^{3} y^{2}) - \log_{b} (z^{4})$ .

$\frac{1}{2} (\log_{b} (x^{3} y^{2}) - \log_{b} (z^{4}))$

Step 4

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

$\frac{1}{2} (\log_{b} (x^{3} y^{2}) - (4 \log_{b} (z)))$

Step 5

Multiply $4$ by $- 1$ .

$\frac{1}{2} (\log_{b} (x^{3} y^{2}) - 4 \log_{b} (z))$

Step 6

Rewrite $\log_{b} (x^{3} y^{2})$ as $\log_{b} (x^{3}) + \log_{b} (y^{2})$ .

$\frac{1}{2} (\log_{b} (x^{3}) + \log_{b} (y^{2}) - 4 \log_{b} (z))$

Step 7

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

$\frac{1}{2} (3 \log_{b} (x) + \log_{b} (y^{2}) - 4 \log_{b} (z))$

Step 8

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

$\frac{1}{2} (3 \log_{b} (x) + 2 \log_{b} (y) - 4 \log_{b} (z))$

Step 9

Apply the distributive property.

$\frac{1}{2} (3 \log_{b} (x)) + \frac{1}{2} (2 \log_{b} (y)) + \frac{1}{2} (- 4 \log_{b} (z))$

Step 10

Simplify.

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Step 10.1

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

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Step 10.1.1

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

$\frac{3}{2} \log_{b} (x) + \frac{1}{2} (2 \log_{b} (y)) + \frac{1}{2} (- 4 \log_{b} (z))$

Step 10.1.2

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

$\frac{3 \log_{b} (x)}{2} + \frac{1}{2} (2 \log_{b} (y)) + \frac{1}{2} (- 4 \log_{b} (z))$

Step 10.2

Cancel the common factor of $2$ .

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Step 10.2.1

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

$\frac{3 \log_{b} (x)}{2} + \frac{1}{2} (2 (\log_{b} (y))) + \frac{1}{2} (- 4 \log_{b} (z))$

Step 10.2.2

Cancel the common factor.

$\frac{3 \log_{b} (x)}{2} + \frac{1}{2} (2 \log_{b} (y)) + \frac{1}{2} (- 4 \log_{b} (z))$

Step 10.2.3

Rewrite the expression.

$\frac{3 \log_{b} (x)}{2} + \log_{b} (y) + \frac{1}{2} (- 4 \log_{b} (z))$

Step 10.3

Cancel the common factor of $2$ .

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Step 10.3.1

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

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

Step 10.3.2

Cancel the common factor.

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

Step 10.3.3

Rewrite the expression.

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