Precalculus Examples

$\log_{5} (\frac{125 x (\sqrt{x + 1})}{{(x - 3)}^{4}})$

Step 1

Rewrite $\log_{5} (\frac{125 x (\sqrt{x + 1})}{{(x - 3)}^{4}})$ as $\log_{5} (125 x (\sqrt{x + 1})) - \log_{5} ({(x - 3)}^{4})$ .

$\log_{5} (125 x (\sqrt{x + 1})) - \log_{5} ({(x - 3)}^{4})$

Step 2

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

$\log_{5} (125 x {(x + 1)}^{\frac{1}{2}}) - \log_{5} ({(x - 3)}^{4})$

Step 3

Expand $\log_{5} ({(x - 3)}^{4})$ by moving $4$ outside the logarithm.

$\log_{5} (125 x {(x + 1)}^{\frac{1}{2}}) - (4 \log_{5} (x - 3))$

Step 4

Multiply $4$ by $- 1$ .

$\log_{5} (125 x {(x + 1)}^{\frac{1}{2}}) - 4 \log_{5} (x - 3)$

Step 5

Rewrite $\log_{5} (125 x {(x + 1)}^{\frac{1}{2}})$ as $\log_{5} (125 x) + \log_{5} ({(x + 1)}^{\frac{1}{2}})$ .

$\log_{5} (125 x) + \log_{5} ({(x + 1)}^{\frac{1}{2}}) - 4 \log_{5} (x - 3)$

Step 6

Rewrite $\log_{5} (125 x)$ as $\log_{5} (125) + \log_{5} (x)$ .

$\log_{5} (125) + \log_{5} (x) + \log_{5} ({(x + 1)}^{\frac{1}{2}}) - 4 \log_{5} (x - 3)$

Step 7

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

$\log_{5} (125) + \log_{5} (x) + \frac{1}{2} \log_{5} (x + 1) - 4 \log_{5} (x - 3)$

Step 8

Logarithm base $5$ of $125$ is $3$ .

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

Step 9

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

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

Step 10

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 11

Combine $3$ and $\frac{2}{2}$ .

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

Multiply $3$ by $2$ .

$\frac{6 + \log_{5} (x + 1)}{2} + \log_{5} (x) - 4 \log_{5} (x - 3)$

Step 14

To write $- 4 \log_{5} (x - 3)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{6 + \log_{5} (x + 1)}{2} - 4 \log_{5} (x - 3) \cdot \frac{2}{2} + \log_{5} (x)$

Step 15

Combine $- 4 \log_{5} (x - 3)$ and $\frac{2}{2}$ .

$\frac{6 + \log_{5} (x + 1)}{2} + \frac{- 4 \log_{5} (x - 3) \cdot 2}{2} + \log_{5} (x)$

Step 16

Combine the numerators over the common denominator.

$\frac{6 + \log_{5} (x + 1) - 4 \log_{5} (x - 3) \cdot 2}{2} + \log_{5} (x)$

Step 17

Multiply $2$ by $- 4$ .

$\frac{6 + \log_{5} (x + 1) - 8 \log_{5} (x - 3)}{2} + \log_{5} (x)$