Precalculus Examples

$\frac{18 \ln (2) - \ln (13)}{3 \ln (2)}$

Step 1

Simplify the numerator.

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Step 1.1

Simplify $18 \ln (2)$ by moving $18$ inside the logarithm.

$\frac{\ln (2^{18}) - \ln (13)}{3 \ln (2)}$

Step 1.2

Raise $2$ to the power of $18$ .

$\frac{\ln (262144) - \ln (13)}{3 \ln (2)}$

Step 1.3

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

$\frac{\ln (\frac{262144}{13})}{3 \ln (2)}$

Step 2

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

$\frac{\ln (\frac{262144}{13})}{\ln (2^{3})}$

Step 3

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

$\frac{\ln (\frac{262144}{13})}{\ln (8)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (\frac{262144}{13})}{\ln (8)}$

Decimal Form:

$4.76652009 \dots$