Precalculus Examples

$\ln (\frac{1}{9^{k}})$

Step 1

Rewrite $\ln (\frac{1}{9^{k}})$ as $\ln (1) - \ln (9^{k})$ .

$\ln (1) - \ln (9^{k})$

Step 2

Expand $\ln (9^{k})$ by moving $k$ outside the logarithm.

$\ln (1) - (k \ln (9))$

Step 3

The natural logarithm of $1$ is $0$ .

$0 - (k \ln (9))$

Step 4

Subtract $k \ln (9)$ from $0$ .

$- (k \ln (9))$

Step 5

Remove parentheses.

$- k \ln (9)$

Step 6

Rewrite $\ln (9)$ as $\ln (3^{2})$ .

$- k \ln (3^{2})$

Step 7

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

$- k (2 \ln (3))$

Step 8

Multiply $2$ by $- 1$ .

$- 2 k \ln (3)$