Precalculus Examples

$\log_{4} (\frac{w z^{5}}{\sqrt{x y^{5}}})$

Step 1

Rewrite $\log_{4} (\frac{w z^{5}}{\sqrt{x y^{5}}})$ as $\log_{4} (w z^{5}) - \log_{4} (\sqrt{x y^{5}})$ .

$\log_{4} (w z^{5}) - \log_{4} (\sqrt{x y^{5}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x y^{5}}$ as ${(x y^{5})}^{\frac{1}{2}}$ .

$\log_{4} (w z^{5}) - \log_{4} ({(x y^{5})}^{\frac{1}{2}})$

Step 3

Expand $\log_{4} ({(x y^{5})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{4} (w z^{5}) - (\frac{1}{2} \log_{4} (x y^{5}))$

Step 4

Combine $\frac{1}{2}$ and $\log_{4} (x y^{5})$ .

$\log_{4} (w z^{5}) - \frac{\log_{4} (x y^{5})}{2}$

Step 5

Rewrite $\log_{4} (w z^{5})$ as $\log_{4} (w) + \log_{4} (z^{5})$ .

$\log_{4} (w) + \log_{4} (z^{5}) - \frac{\log_{4} (x y^{5})}{2}$

Step 6

Expand $\log_{4} (z^{5})$ by moving $5$ outside the logarithm.

$\log_{4} (w) + 5 \log_{4} (z) - \frac{\log_{4} (x y^{5})}{2}$

Step 7

Rewrite $\log_{4} (x y^{5})$ as $\log_{4} (x) + \log_{4} (y^{5})$ .

$\log_{4} (w) + 5 \log_{4} (z) - \frac{\log_{4} (x) + \log_{4} (y^{5})}{2}$

Step 8

Expand $\log_{4} (y^{5})$ by moving $5$ outside the logarithm.

$\log_{4} (w) + 5 \log_{4} (z) - \frac{\log_{4} (x) + 5 \log_{4} (y)}{2}$

Step 9

To write $\log_{4} (w)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5 \log_{4} (z) + \log_{4} (w) \cdot \frac{2}{2} - \frac{\log_{4} (x) + 5 \log_{4} (y)}{2}$

Step 10

Combine $\log_{4} (w)$ and $\frac{2}{2}$ .

$5 \log_{4} (z) + \frac{\log_{4} (w) \cdot 2}{2} - \frac{\log_{4} (x) + 5 \log_{4} (y)}{2}$

Step 11

Combine the numerators over the common denominator.

$5 \log_{4} (z) + \frac{\log_{4} (w) \cdot 2 - (\log_{4} (x) + 5 \log_{4} (y))}{2}$

Step 12

Simplify the numerator.

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Step 12.1

Move $2$ to the left of $\log_{4} (w)$ .

$5 \log_{4} (z) + \frac{2 \cdot \log_{4} (w) - (\log_{4} (x) + 5 \log_{4} (y))}{2}$

Step 12.2

Apply the distributive property.

$5 \log_{4} (z) + \frac{2 \log_{4} (w) - \log_{4} (x) - (5 \log_{4} (y))}{2}$

Step 12.3

Multiply $5$ by $- 1$ .

$5 \log_{4} (z) + \frac{2 \log_{4} (w) - \log_{4} (x) - 5 \log_{4} (y)}{2}$

Step 13

To write $5 \log_{4} (z)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5 \log_{4} (z) \cdot \frac{2}{2} + \frac{2 \log_{4} (w) - \log_{4} (x) - 5 \log_{4} (y)}{2}$

Step 14

Combine $5 \log_{4} (z)$ and $\frac{2}{2}$ .

$\frac{5 \log_{4} (z) \cdot 2}{2} + \frac{2 \log_{4} (w) - \log_{4} (x) - 5 \log_{4} (y)}{2}$

Step 15

Combine the numerators over the common denominator.

$\frac{5 \log_{4} (z) \cdot 2 + 2 \log_{4} (w) - \log_{4} (x) - 5 \log_{4} (y)}{2}$

Step 16

Multiply $2$ by $5$ .

$\frac{10 \log_{4} (z) + 2 \log_{4} (w) - \log_{4} (x) - 5 \log_{4} (y)}{2}$