Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Simplify $2 \ln (2)$ by moving $2$ inside the logarithm.

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

Step 1.2

Raise $2$ to the power of $2$ .

$\frac{\ln (4) - \ln (3)}{3 \ln (2) - \ln (3)}$

Step 1.3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{\ln (\frac{4}{3})}{3 \ln (2) - \ln (3)}$

Step 2

Simplify the denominator.

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Step 2.1

Simplify $3 \ln (2)$ by moving $3$ inside the logarithm.

$\frac{\ln (\frac{4}{3})}{\ln (2^{3}) - \ln (3)}$

Step 2.2

Raise $2$ to the power of $3$ .

$\frac{\ln (\frac{4}{3})}{\ln (8) - \ln (3)}$

Step 2.3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{\ln (\frac{4}{3})}{\ln (\frac{8}{3})}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (\frac{4}{3})}{\ln (\frac{8}{3})}$

Decimal Form:

$0.29330494 \dots$