Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $3$ by $- 1$ .

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

Step 6

Rewrite $\log_{b} (x^{4} y^{2})$ as $\log_{b} (x^{4}) + \log_{b} (y^{2})$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Apply the distributive property.

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

Step 10

Simplify.

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Step 10.1

Cancel the common factor of $2$ .

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Step 10.1.1

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

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

Step 10.1.2

Cancel the common factor.

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

Step 10.1.3

Rewrite the expression.

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

Step 10.2

Cancel the common factor of $2$ .

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Step 10.2.1

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

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

Step 10.2.2

Cancel the common factor.

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

Step 10.2.3

Rewrite the expression.

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

Step 10.3

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

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Step 10.3.1

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

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

Step 10.3.2

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

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

Step 11

Move the negative in front of the fraction.

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