Precalculus Examples

$\log_{3} (\frac{3 x}{\sqrt{x - 1}})$

Step 1

Rewrite $\log_{3} (\frac{3 x}{\sqrt{x - 1}})$ as $\log_{3} (3 x) - \log_{3} (\sqrt{x - 1})$ .

$\log_{3} (3 x) - \log_{3} (\sqrt{x - 1})$

Step 2

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

$\log_{3} (3 x) - \log_{3} ({(x - 1)}^{\frac{1}{2}})$

Step 3

Expand $\log_{3} ({(x - 1)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 4

Combine $\frac{1}{2}$ and $\log_{3} (x - 1)$ .

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

Step 5

Rewrite $\log_{3} (3 x)$ as $\log_{3} (3) + \log_{3} (x)$ .

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

Step 6

Logarithm base $3$ of $3$ is $1$ .

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

Step 7

Write $1$ as a fraction with a common denominator.

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

Step 8

Combine the numerators over the common denominator.

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