Precalculus Examples

$\frac{10 \ln (2)}{2 \ln (2) - 9 \ln (3)}$

Step 1

Simplify $10 \ln (2)$ by moving $10$ inside the logarithm.

$\frac{\ln (2^{10})}{2 \ln (2) - 9 \ln (3)}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{\ln (2^{10})}{\ln (2^{2}) - 9 \ln (3)}$

Step 2.2

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

$\frac{\ln (2^{10})}{\ln (4) - 9 \ln (3)}$

Step 2.3

Simplify $- 9 \ln (3)$ by moving $9$ inside the logarithm.

$\frac{\ln (2^{10})}{\ln (4) - \ln (3^{9})}$

Step 2.4

Raise $3$ to the power of $9$ .

$\frac{\ln (2^{10})}{\ln (4) - \ln (19683)}$

Step 2.5

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

$\frac{\ln (2^{10})}{\ln (\frac{4}{19683})}$

Step 3

Raise $2$ to the power of $10$ .

$\frac{\ln (1024)}{\ln (\frac{4}{19683})}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (1024)}{\ln (\frac{4}{19683})}$

Decimal Form:

$- 0.81535060 \dots$