Precalculus Examples

$\frac{5 (\ln (\frac{1}{3}))}{\ln (\frac{11}{15})}$

Step 1

Rewrite $\ln (\frac{1}{3})$ as $\ln (1) - \ln (3)$ .

$\frac{5 (\ln (1) - \ln (3))}{\ln (\frac{11}{15})}$

Step 2

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

$\frac{5 (0 - \ln (3))}{\ln (\frac{11}{15})}$

Step 3

Subtract $\ln (3)$ from $0$ .

$\frac{5 (- \ln (3))}{\ln (\frac{11}{15})}$

Step 4

Rewrite $\ln (\frac{11}{15})$ as $\ln (11) - \ln (15)$ .

$\frac{5 (- \ln (3))}{\ln (11) - \ln (15)}$

Step 5

Rewrite $\ln (15)$ as $\ln (3 (5))$ .

$\frac{5 (- \ln (3))}{\ln (11) - \ln (3 (5))}$

Step 6

Rewrite $\ln (3 (5))$ as $\ln (3) + \ln (5)$ .

$\frac{5 (- \ln (3))}{\ln (11) - (\ln (3) + \ln (5))}$

Step 7

Apply the distributive property.

$\frac{5 (- \ln (3))}{\ln (11) - \ln (3) - \ln (5)}$

Step 8

Multiply $- 1$ by $5$ .

$\frac{- 5 \ln (3)}{\ln (11) - \ln (3) - \ln (5)}$

Step 9

Move the negative in front of the fraction.

$- \frac{5 \ln (3)}{\ln (11) - \ln (3) - \ln (5)}$