Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Simplify $- 2 \ln (5)$ by moving $2$ inside the logarithm.

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

Step 1.2

Raise $5$ to the power of $2$ .

$\frac{- \ln (3) - \ln (25)}{\ln (3) - \ln (5)}$

Step 1.3

Rewrite $- \ln (3) - \ln (25)$ in a factored form.

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Step 1.3.1

Factor $- 1$ out of $- \ln (3) - \ln (25)$ .

$\frac{- (\ln (3) + \ln (25))}{\ln (3) - \ln (5)}$

Step 1.3.2

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

$\frac{- \ln (3 \cdot 25)}{\ln (3) - \ln (5)}$

Step 1.4

Multiply $3$ by $25$ .

$\frac{- \ln (75)}{\ln (3) - \ln (5)}$

Step 2

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

$\frac{- \ln (75)}{\ln (\frac{3}{5})}$

Step 3

Move the negative in front of the fraction.

$- \frac{\ln (75)}{\ln (\frac{3}{5})}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{\ln (75)}{\ln (\frac{3}{5})}$

Decimal Form:

$8.45198030 \dots$