Precalculus Examples

$\log_{5} (\frac{\sqrt{x + 2}}{y^{3} {(z - 1)}^{4}})$

Step 1

Rewrite $\log_{5} (\frac{\sqrt{x + 2}}{y^{3} {(z - 1)}^{4}})$ as $\log_{5} (\sqrt{x + 2}) - \log_{5} (y^{3} {(z - 1)}^{4})$ .

$\log_{5} (\sqrt{x + 2}) - \log_{5} (y^{3} {(z - 1)}^{4})$

Step 2

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

$\log_{5} ({(x + 2)}^{\frac{1}{2}}) - \log_{5} (y^{3} {(z - 1)}^{4})$

Step 3

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

$\frac{1}{2} \log_{5} (x + 2) - \log_{5} (y^{3} {(z - 1)}^{4})$

Step 4

Rewrite $\log_{5} (y^{3} {(z - 1)}^{4})$ as $\log_{5} (y^{3}) + \log_{5} ({(z - 1)}^{4})$ .

$\frac{1}{2} \log_{5} (x + 2) - (\log_{5} (y^{3}) + \log_{5} ({(z - 1)}^{4}))$

Step 5

Expand $\log_{5} (y^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{2} \log_{5} (x + 2) - (3 \log_{5} (y) + \log_{5} ({(z - 1)}^{4}))$

Step 6

Expand $\log_{5} ({(z - 1)}^{4})$ by moving $4$ outside the logarithm.

$\frac{1}{2} \log_{5} (x + 2) - (3 \log_{5} (y) + 4 \log_{5} (z - 1))$

Step 7

Simplify each term.

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Step 7.1

Combine $\frac{1}{2}$ and $\log_{5} (x + 2)$ .

$\frac{\log_{5} (x + 2)}{2} - (3 \log_{5} (y) + 4 \log_{5} (z - 1))$

Step 7.2

Apply the distributive property.

$\frac{\log_{5} (x + 2)}{2} - (3 \log_{5} (y)) - (4 \log_{5} (z - 1))$

Step 7.3

Multiply $3$ by $- 1$ .

$\frac{\log_{5} (x + 2)}{2} - 3 \log_{5} (y) - (4 \log_{5} (z - 1))$

Step 7.4

Multiply $4$ by $- 1$ .

$\frac{\log_{5} (x + 2)}{2} - 3 \log_{5} (y) - 4 \log_{5} (z - 1)$

Step 8

To write $- 3 \log_{5} (y)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\log_{5} (x + 2)}{2} - 3 \log_{5} (y) \cdot \frac{2}{2} - 4 \log_{5} (z - 1)$

Step 9

Combine $- 3 \log_{5} (y)$ and $\frac{2}{2}$ .

$\frac{\log_{5} (x + 2)}{2} + \frac{- 3 \log_{5} (y) \cdot 2}{2} - 4 \log_{5} (z - 1)$

Step 10

Combine the numerators over the common denominator.

$\frac{\log_{5} (x + 2) - 3 \log_{5} (y) \cdot 2}{2} - 4 \log_{5} (z - 1)$

Step 11

Multiply $2$ by $- 3$ .

$\frac{\log_{5} (x + 2) - 6 \log_{5} (y)}{2} - 4 \log_{5} (z - 1)$

Step 12

To write $- 4 \log_{5} (z - 1)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\log_{5} (x + 2) - 6 \log_{5} (y)}{2} - 4 \log_{5} (z - 1) \cdot \frac{2}{2}$

Step 13

Combine $- 4 \log_{5} (z - 1)$ and $\frac{2}{2}$ .

$\frac{\log_{5} (x + 2) - 6 \log_{5} (y)}{2} + \frac{- 4 \log_{5} (z - 1) \cdot 2}{2}$

Step 14

Combine the numerators over the common denominator.

$\frac{\log_{5} (x + 2) - 6 \log_{5} (y) - 4 \log_{5} (z - 1) \cdot 2}{2}$

Step 15

Multiply $2$ by $- 4$ .

$\frac{\log_{5} (x + 2) - 6 \log_{5} (y) - 8 \log_{5} (z - 1)}{2}$