Precalculus Examples

$\frac{- 2 \ln (3) - 3 \ln (5)}{4 \ln (3) - \ln (5)}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{- \ln (3^{2}) - 3 \ln (5)}{4 \ln (3) - \ln (5)}$

Step 1.2

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

$\frac{- \ln (9) - 3 \ln (5)}{4 \ln (3) - \ln (5)}$

Step 1.3

Simplify $- 3 \ln (5)$ by moving $3$ inside the logarithm.

$\frac{- \ln (9) - \ln (5^{3})}{4 \ln (3) - \ln (5)}$

Step 1.4

Raise $5$ to the power of $3$ .

$\frac{- \ln (9) - \ln (125)}{4 \ln (3) - \ln (5)}$

Step 1.5

Rewrite $- \ln (9) - \ln (125)$ in a factored form.

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Step 1.5.1

Factor $- 1$ out of $- \ln (9) - \ln (125)$ .

$\frac{- (\ln (9) + \ln (125))}{4 \ln (3) - \ln (5)}$

Step 1.5.2

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

$\frac{- \ln (9 \cdot 125)}{4 \ln (3) - \ln (5)}$

Step 1.6

Multiply $9$ by $125$ .

$\frac{- \ln (1125)}{4 \ln (3) - \ln (5)}$

Step 2

Simplify the denominator.

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Step 2.1

Simplify $4 \ln (3)$ by moving $4$ inside the logarithm.

$\frac{- \ln (1125)}{\ln (3^{4}) - \ln (5)}$

Step 2.2

Raise $3$ to the power of $4$ .

$\frac{- \ln (1125)}{\ln (81) - \ln (5)}$

Step 2.3

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

$\frac{- \ln (1125)}{\ln (\frac{81}{5})}$

Step 3

Move the negative in front of the fraction.

$- \frac{\ln (1125)}{\ln (\frac{81}{5})}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{\ln (1125)}{\ln (\frac{81}{5})}$

Decimal Form:

$- 2.52262475 \dots$