Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 1.2

Multiply $3$ by $2$ .

$\frac{2 \ln (6)}{\ln (3) - 4 \ln (2)}$

Step 2

Simplify the denominator.

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Step 2.1

Simplify $- 4 \ln (2)$ by moving $4$ inside the logarithm.

$\frac{2 \ln (6)}{\ln (3) - \ln (2^{4})}$

Step 2.2

Raise $2$ to the power of $4$ .

$\frac{2 \ln (6)}{\ln (3) - \ln (16)}$

Step 2.3

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

$\frac{2 \ln (6)}{\ln (\frac{3}{16})}$

Step 3

Simplify $2 \ln (6)$ by moving $2$ inside the logarithm.

$\frac{\ln (6^{2})}{\ln (\frac{3}{16})}$

Step 4

Raise $6$ to the power of $2$ .

$\frac{\ln (36)}{\ln (\frac{3}{16})}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (36)}{\ln (\frac{3}{16})}$

Decimal Form:

$- 2.14072245 \dots$