Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

$\frac{\ln (3125) - 4 \ln (4)}{\ln (4) - 2 \ln (5)}$

Step 1.3

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

$\frac{\ln (3125) - \ln (4^{4})}{\ln (4) - 2 \ln (5)}$

Step 1.4

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

$\frac{\ln (3125) - \ln (256)}{\ln (4) - 2 \ln (5)}$

Step 1.5

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

$\frac{\ln (\frac{3125}{256})}{\ln (4) - 2 \ln (5)}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{\ln (\frac{3125}{256})}{\ln (4) - \ln (5^{2})}$

Step 2.2

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

$\frac{\ln (\frac{3125}{256})}{\ln (4) - \ln (25)}$

Step 2.3

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

$\frac{\ln (\frac{3125}{256})}{\ln (\frac{4}{25})}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (\frac{3125}{256})}{\ln (\frac{4}{25})}$

Decimal Form:

$- 1.36529380 \dots$