Precalculus Examples

$\frac{3 \ln (5) + 3 \ln (3)}{2 \ln (3) - \ln (5)}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $3$ out of $3 \ln (5) + 3 \ln (3)$ .

$\frac{3 (\ln (5) + \ln (3))}{2 \ln (3) - \ln (5)}$

Step 1.2

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

$\frac{3 \ln (5 \cdot 3)}{2 \ln (3) - \ln (5)}$

Step 1.3

Multiply $5$ by $3$ .

$\frac{3 \ln (15)}{2 \ln (3) - \ln (5)}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{3 \ln (15)}{\ln (3^{2}) - \ln (5)}$

Step 2.2

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

$\frac{3 \ln (15)}{\ln (9) - \ln (5)}$

Step 2.3

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

$\frac{3 \ln (15)}{\ln (\frac{9}{5})}$

Step 3

Simplify $3 \ln (15)$ by moving $3$ inside the logarithm.

$\frac{\ln (15^{3})}{\ln (\frac{9}{5})}$

Step 4

Raise $15$ to the power of $3$ .

$\frac{\ln (3375)}{\ln (\frac{9}{5})}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (3375)}{\ln (\frac{9}{5})}$

Decimal Form:

$13.82159733 \dots$