Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

$\frac{\ln (4 \cdot 27)}{\ln ({(3 \cdot 5)}^{- 2})}$

Step 1.3

Multiply $4$ by $27$ .

$\frac{\ln (108)}{\ln ({(3 \cdot 5)}^{- 2})}$

Step 2

Simplify the denominator.

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Step 2.1

Multiply $3$ by $5$ .

$\frac{\ln (108)}{\ln (15^{- 2})}$

Step 2.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\ln (108)}{\ln (\frac{1}{15^{2}})}$

Step 2.3

Raise $15$ to the power of $2$ .

$\frac{\ln (108)}{\ln (\frac{1}{225})}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (108)}{\ln (\frac{1}{225})}$

Decimal Form:

$- 0.86448383 \dots$