Precalculus Examples

$\ln (\frac{\sqrt{81 - x^{2}}}{(x - 7) {(x + 7)}^{\frac{3}{2}}})$

Step 1

Rewrite $\ln (\frac{\sqrt{81 - x^{2}}}{(x - 7) {(x + 7)}^{\frac{3}{2}}})$ as $\ln (\sqrt{81 - x^{2}}) - \ln ((x - 7) {(x + 7)}^{\frac{3}{2}})$ .

$\ln (\sqrt{81 - x^{2}}) - \ln ((x - 7) {(x + 7)}^{\frac{3}{2}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{81 - x^{2}}$ as ${(81 - x^{2})}^{\frac{1}{2}}$ .

$\ln ({(81 - x^{2})}^{\frac{1}{2}}) - \ln ((x - 7) {(x + 7)}^{\frac{3}{2}})$

Step 3

Expand $\ln ({(81 - x^{2})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \ln (81 - x^{2}) - \ln ((x - 7) {(x + 7)}^{\frac{3}{2}})$

Step 4

Rewrite $\ln ((x - 7) {(x + 7)}^{\frac{3}{2}})$ as $\ln (x - 7) + \ln ({(x + 7)}^{\frac{3}{2}})$ .

$\frac{1}{2} \ln (81 - x^{2}) - (\ln (x - 7) + \ln ({(x + 7)}^{\frac{3}{2}}))$

Step 5

Expand $\ln ({(x + 7)}^{\frac{3}{2}})$ by moving $\frac{3}{2}$ outside the logarithm.

$\frac{1}{2} \ln (81 - x^{2}) - (\ln (x - 7) + \frac{3}{2} \ln (x + 7))$

Step 6

Simplify each term.

Tap for more steps...

Step 6.1

Combine $\frac{1}{2}$ and $\ln (81 - x^{2})$ .

$\frac{\ln (81 - x^{2})}{2} - (\ln (x - 7) + \frac{3}{2} \ln (x + 7))$

Step 6.2

Combine $\frac{3}{2}$ and $\ln (x + 7)$ .

$\frac{\ln (81 - x^{2})}{2} - (\ln (x - 7) + \frac{3 \ln (x + 7)}{2})$

Step 6.3

To write $\ln (x - 7)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\ln (81 - x^{2})}{2} - (\ln (x - 7) \cdot \frac{2}{2} + \frac{3 \ln (x + 7)}{2})$

Step 6.4

Combine $\ln (x - 7)$ and $\frac{2}{2}$ .

$\frac{\ln (81 - x^{2})}{2} - (\frac{\ln (x - 7) \cdot 2}{2} + \frac{3 \ln (x + 7)}{2})$

Step 6.5

Combine the numerators over the common denominator.

$\frac{\ln (81 - x^{2})}{2} - \frac{\ln (x - 7) \cdot 2 + 3 \ln (x + 7)}{2}$

Step 6.6

Move $2$ to the left of $\ln (x - 7)$ .

$\frac{\ln (81 - x^{2})}{2} - \frac{2 \ln (x - 7) + 3 \ln (x + 7)}{2}$

Step 7

Combine the numerators over the common denominator.

$\frac{\ln (81 - x^{2}) - (2 \ln (x - 7) + 3 \ln (x + 7))}{2}$

Step 8

Simplify each term.

Tap for more steps...

Step 8.1

Apply the distributive property.

$\frac{\ln (81 - x^{2}) - (2 \ln (x - 7)) - (3 \ln (x + 7))}{2}$

Step 8.2

Multiply $2$ by $- 1$ .

$\frac{\ln (81 - x^{2}) - 2 \ln (x - 7) - (3 \ln (x + 7))}{2}$

Step 8.3

Multiply $3$ by $- 1$ .

$\frac{\ln (81 - x^{2}) - 2 \ln (x - 7) - 3 \ln (x + 7)}{2}$