Precalculus Examples

$\frac{\log (x^{2})}{\log (y \sqrt{z})}$

Step 1

Expand $\log (x^{2})$ by moving $2$ outside the logarithm.

$\frac{2 \log (x)}{\log (y \sqrt{z})}$

Step 2

Rewrite $\log (y \sqrt{z})$ as $\log (y) + \log (\sqrt{z})$ .

$\frac{2 \log (x)}{\log (y) + \log (\sqrt{z})}$

Step 3

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

$\frac{2 \log (x)}{\log (y) + \log (z^{\frac{1}{2}})}$

Step 4

Expand $\log (z^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{2 \log (x)}{\log (y) + \frac{1}{2} \log (z)}$

Step 5

Simplify the denominator.

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Step 5.1

Combine $\frac{1}{2}$ and $\log (z)$ .

$\frac{2 \log (x)}{\log (y) + \frac{\log (z)}{2}}$

Step 5.2

To write $\log (y)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 \log (x)}{\log (y) \cdot \frac{2}{2} + \frac{\log (z)}{2}}$

Step 5.3

Combine $\log (y)$ and $\frac{2}{2}$ .

$\frac{2 \log (x)}{\frac{\log (y) \cdot 2}{2} + \frac{\log (z)}{2}}$

Step 5.4

Combine the numerators over the common denominator.

$\frac{2 \log (x)}{\frac{\log (y) \cdot 2 + \log (z)}{2}}$

Step 5.5

Move $2$ to the left of $\log (y)$ .

$\frac{2 \log (x)}{\frac{2 \log (y) + \log (z)}{2}}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$2 \log (x) \frac{2}{2 \log (y) + \log (z)}$

Step 7

Multiply $2 \log (x) \frac{2}{2 \log (y) + \log (z)}$ .

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Step 7.1

Combine $\frac{2}{2 \log (y) + \log (z)}$ and $2$ .

$\frac{2 \cdot 2}{2 \log (y) + \log (z)} \log (x)$

Step 7.2

Multiply $2$ by $2$ .

$\frac{4}{2 \log (y) + \log (z)} \log (x)$

Step 7.3

Combine $\frac{4}{2 \log (y) + \log (z)}$ and $\log (x)$ .

$\frac{4 \log (x)}{2 \log (y) + \log (z)}$