Precalculus Examples

$\frac{\ln (2)}{12 \cdot \ln (1 + \frac{4}{12})}$

Step 1

Cancel the common factor of $4$ and $12$ .

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Step 1.1

Factor $4$ out of $4$ .

$\frac{\ln (2)}{12 \cdot \ln (1 + \frac{4 (1)}{12})}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $4$ out of $12$ .

$\frac{\ln (2)}{12 \cdot \ln (1 + \frac{4 \cdot 1}{4 \cdot 3})}$

Step 1.2.2

Cancel the common factor.

$\frac{\ln (2)}{12 \cdot \ln (1 + \frac{4 \cdot 1}{4 \cdot 3})}$

Step 1.2.3

Rewrite the expression.

$\frac{\ln (2)}{12 \cdot \ln (1 + \frac{1}{3})}$

Step 2

Simplify $12 \ln (1 + \frac{1}{3})$ by moving $12$ inside the logarithm.

$\frac{\ln (2)}{\ln ({(1 + \frac{1}{3})}^{12})}$

Step 3

Simplify the denominator.

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Step 3.1

Write $1$ as a fraction with a common denominator.

$\frac{\ln (2)}{\ln ({(\frac{3}{3} + \frac{1}{3})}^{12})}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{\ln (2)}{\ln ({(\frac{3 + 1}{3})}^{12})}$

Step 3.3

Add $3$ and $1$ .

$\frac{\ln (2)}{\ln ({(\frac{4}{3})}^{12})}$

Step 3.4

Apply the product rule to $\frac{4}{3}$ .

$\frac{\ln (2)}{\ln (\frac{4^{12}}{3^{12}})}$

Step 3.5

Raise $4$ to the power of $12$ .

$\frac{\ln (2)}{\ln (\frac{16777216}{3^{12}})}$

Step 3.6

Raise $3$ to the power of $12$ .

$\frac{\ln (2)}{\ln (\frac{16777216}{531441})}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (2)}{\ln (\frac{16777216}{531441})}$

Decimal Form:

$0.20078506 \dots$