Precalculus Examples

$\log (\frac{x^{3 \sqrt{x + 1}}}{{(x - 2)}^{2}})$

Step 1

Rewrite $\log (\frac{x^{3 \sqrt{x + 1}}}{{(x - 2)}^{2}})$ as $\log (x^{3 \sqrt{x + 1}}) - \log ({(x - 2)}^{2})$ .

$\log (x^{3 \sqrt{x + 1}}) - \log ({(x - 2)}^{2})$

Step 2

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

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

Step 3

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

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

Step 4

Expand $\log ({(x - 2)}^{2})$ by moving $2$ outside the logarithm.

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

Step 5

Multiply $2$ by $- 1$ .

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