Precalculus Examples

$\log_{x} (\sqrt{\frac{x^{4} y^{6}}{z^{8}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x^{4} y^{6}}{z^{8}}}$ as ${(\frac{x^{4} y^{6}}{z^{8}})}^{\frac{1}{2}}$ .

$\log_{x} ({(\frac{x^{4} y^{6}}{z^{8}})}^{\frac{1}{2}})$

Step 2

Expand $\log_{x} ({(\frac{x^{4} y^{6}}{z^{8}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{x} (\frac{x^{4} y^{6}}{z^{8}})$

Step 3

Rewrite $\log_{x} (\frac{x^{4} y^{6}}{z^{8}})$ as $\log_{x} (x^{4} y^{6}) - \log_{x} (z^{8})$ .

$\frac{1}{2} (\log_{x} (x^{4} y^{6}) - \log_{x} (z^{8}))$

Step 4

Expand $\log_{x} (z^{8})$ by moving $8$ outside the logarithm.

$\frac{1}{2} (\log_{x} (x^{4} y^{6}) - (8 \log_{x} (z)))$

Step 5

Multiply $8$ by $- 1$ .

$\frac{1}{2} (\log_{x} (x^{4} y^{6}) - 8 \log_{x} (z))$

Step 6

Rewrite $\log_{x} (x^{4} y^{6})$ as $\log_{x} (x^{4}) + \log_{x} (y^{6})$ .

$\frac{1}{2} (\log_{x} (x^{4}) + \log_{x} (y^{6}) - 8 \log_{x} (z))$

Step 7

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

$\frac{1}{2} (4 \log_{x} (x) + \log_{x} (y^{6}) - 8 \log_{x} (z))$

Step 8

Expand $\log_{x} (y^{6})$ by moving $6$ outside the logarithm.

$\frac{1}{2} (4 \log_{x} (x) + 6 \log_{x} (y) - 8 \log_{x} (z))$

Step 9

Logarithm base $x$ of $x$ is $1$ .

$\frac{1}{2} (4 \cdot 1 + 6 \log_{x} (y) - 8 \log_{x} (z))$

Step 10

Multiply $4$ by $1$ .

$\frac{1}{2} (4 + 6 \log_{x} (y) - 8 \log_{x} (z))$

Step 11

Apply the distributive property.

$\frac{1}{2} \cdot 4 + \frac{1}{2} (6 \log_{x} (y)) + \frac{1}{2} (- 8 \log_{x} (z))$

Step 12

Simplify.

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Step 12.1

Cancel the common factor of $2$ .

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Step 12.1.1

Factor $2$ out of $4$ .

$\frac{1}{2} \cdot (2 (2)) + \frac{1}{2} (6 \log_{x} (y)) + \frac{1}{2} (- 8 \log_{x} (z))$

Step 12.1.2

Cancel the common factor.

$\frac{1}{2} \cdot (2 \cdot 2) + \frac{1}{2} (6 \log_{x} (y)) + \frac{1}{2} (- 8 \log_{x} (z))$

Step 12.1.3

Rewrite the expression.

$2 + \frac{1}{2} (6 \log_{x} (y)) + \frac{1}{2} (- 8 \log_{x} (z))$

Step 12.2

Cancel the common factor of $2$ .

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Step 12.2.1

Factor $2$ out of $6 \log_{x} (y)$ .

$2 + \frac{1}{2} (2 (3 \log_{x} (y))) + \frac{1}{2} (- 8 \log_{x} (z))$

Step 12.2.2

Cancel the common factor.

$2 + \frac{1}{2} (2 (3 \log_{x} (y))) + \frac{1}{2} (- 8 \log_{x} (z))$

Step 12.2.3

Rewrite the expression.

$2 + 3 \log_{x} (y) + \frac{1}{2} (- 8 \log_{x} (z))$

Step 12.3

Cancel the common factor of $2$ .

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Step 12.3.1

Factor $2$ out of $- 8 \log_{x} (z)$ .

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

Step 12.3.2

Cancel the common factor.

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

Step 12.3.3

Rewrite the expression.

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