Precalculus Examples

$\frac{\ln (27) + \ln (9)}{\log (27) + \log (9)}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{\ln (27 \cdot 9)}{\log (27) + \log (9)}$

Step 1.2

Multiply $27$ by $9$ .

$\frac{\ln (243)}{\log (27) + \log (9)}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{\ln (243)}{\log (27 \cdot 9)}$

Step 2.2

Multiply $27$ by $9$ .

$\frac{\ln (243)}{\log (243)}$

Step 3

Rewrite $\ln (243)$ as $\ln (3^{5})$ .

$\frac{\ln (3^{5})}{\log (243)}$

Step 4

Expand $\ln (3^{5})$ by moving $5$ outside the logarithm.

$\frac{5 \ln (3)}{\log (243)}$

Step 5

Rewrite $\log (243)$ as $\log (3^{5})$ .

$\frac{5 \ln (3)}{\log (3^{5})}$

Step 6

Expand $\log (3^{5})$ by moving $5$ outside the logarithm.

$\frac{5 \ln (3)}{5 \log (3)}$

Step 7

Cancel the common factor of $5$ .

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Step 7.1

Cancel the common factor.

$\frac{5 \ln (3)}{5 \log (3)}$

Step 7.2

Rewrite the expression.

$\frac{\ln (3)}{\log (3)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (3)}{\log (3)}$

Decimal Form:

$2.30258509 \dots$