Precalculus Examples

$\log_{5} (\sqrt[3]{\frac{x^{2} + 1}{x^{2 - 1}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{x^{2} + 1}{x^{2 - 1}}}$ as ${(\frac{x^{2} + 1}{x^{2 - 1}})}^{\frac{1}{3}}$ .

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

Step 2

Subtract $1$ from $2$ .

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

Step 3

Expand $\log_{5} ({(\frac{x^{2} + 1}{x^{1}})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

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

Step 4

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

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

Step 5

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

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

Step 6

Multiply $\log_{5} (x)$ by $1$ .

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

Step 7

Apply the distributive property.

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

Step 8

Combine $\frac{1}{3}$ and $\log_{5} (x^{2} + 1)$ .

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

Step 9

Combine $\frac{1}{3}$ and $\log_{5} (x)$ .

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

Step 10

Combine the numerators over the common denominator.

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