Precalculus Examples

$\frac{\log (\sqrt{x} \cdot (5 \sqrt{y}))}{z^{14}}$

Step 1

Rewrite $\log (\sqrt{x} \cdot (5 \sqrt{y}))$ as $\log (\sqrt{x}) + \log (5 \sqrt{y})$ .

$\frac{\log (\sqrt{x}) + \log (5 \sqrt{y})}{z^{14}}$

Step 2

Rewrite $\log (5 \sqrt{y})$ as $\log (5) + \log (\sqrt{y})$ .

$\frac{\log (\sqrt{x}) + \log (5) + \log (\sqrt{y})}{z^{14}}$

Step 3

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

$\frac{\log (x^{\frac{1}{2}}) + \log (5) + \log (\sqrt{y})}{z^{14}}$

Step 4

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

$\frac{\log (x^{\frac{1}{2}}) + \log (5) + \log (y^{\frac{1}{2}})}{z^{14}}$

Step 5

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

$\frac{\frac{1}{2} \log (x) + \log (5) + \log (y^{\frac{1}{2}})}{z^{14}}$

Step 6

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

$\frac{\frac{1}{2} \log (x) + \log (5) + \frac{1}{2} \log (y)}{z^{14}}$

Step 7

Simplify the numerator.

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Step 7.1

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

$\frac{\frac{\log (x)}{2} + \log (5) + \frac{1}{2} \log (y)}{z^{14}}$

Step 7.2

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

$\frac{\frac{\log (x)}{2} + \log (5) + \frac{\log (y)}{2}}{z^{14}}$

Step 7.3

To write $\log (5)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{\log (x)}{2} + \log (5) \cdot \frac{2}{2} + \frac{\log (y)}{2}}{z^{14}}$

Step 7.4

Combine $\log (5)$ and $\frac{2}{2}$ .

$\frac{\frac{\log (x)}{2} + \frac{\log (5) \cdot 2}{2} + \frac{\log (y)}{2}}{z^{14}}$

Step 7.5

Combine the numerators over the common denominator.

$\frac{\frac{\log (x) + \log (5) \cdot 2}{2} + \frac{\log (y)}{2}}{z^{14}}$

Step 7.6

Combine the numerators over the common denominator.

$\frac{\frac{\log (x) + \log (5) \cdot 2 + \log (y)}{2}}{z^{14}}$

Step 7.7

Move $2$ to the left of $\log (5)$ .

$\frac{\frac{\log (x) + 2 \log (5) + \log (y)}{2}}{z^{14}}$

Step 8

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log (x) + 2 \log (5) + \log (y)}{2} \cdot \frac{1}{z^{14}}$

Step 9

Multiply $\frac{\log (x) + 2 \log (5) + \log (y)}{2}$ by $\frac{1}{z^{14}}$ .

$\frac{\log (x) + 2 \log (5) + \log (y)}{2 z^{14}}$