Precalculus Examples

$\frac{\ln (9)}{\ln (\frac{27}{8})}$

Step 1

Rewrite $\ln (\frac{27}{8})$ as $\ln (27) - \ln (8)$ .

$\frac{\ln (9)}{\ln (27) - \ln (8)}$

Step 2

Rewrite $\ln (8)$ as $\ln (2^{3})$ .

$\frac{\ln (9)}{\ln (27) - \ln (2^{3})}$

Step 3

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

$\frac{\ln (9)}{\ln (27) - (3 \ln (2))}$

Step 4

Multiply $3$ by $- 1$ .

$\frac{\ln (9)}{\ln (27) - 3 \ln (2)}$

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{\ln (3^{2})}{\ln (27) - 3 \ln (2)}$

Step 5.2

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

$\frac{2 \ln (3)}{\ln (27) - 3 \ln (2)}$

Step 6

Simplify the denominator.

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Step 6.1

Rewrite $\ln (27)$ as $\ln (3^{3})$ .

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

Step 6.2

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

$\frac{2 \ln (3)}{3 \ln (3) - 3 \ln (2)}$