Precalculus Examples

$\frac{1}{3} \cdot (\frac{\ln (\frac{11}{4})}{\ln (2)} + 2)$

Step 1

To write $2$ as a fraction with a common denominator, multiply by $\frac{\ln (2)}{\ln (2)}$ .

$\frac{1}{3} \cdot (\frac{\ln (\frac{11}{4})}{\ln (2)} + \frac{2 \ln (2)}{\ln (2)})$

Step 2

Combine the numerators over the common denominator.

$\frac{1}{3} \cdot \frac{\ln (\frac{11}{4}) + 2 \ln (2)}{\ln (2)}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{1}{3} \cdot \frac{\ln (\frac{11}{4}) + \ln (2^{2})}{\ln (2)}$

Step 3.2

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

$\frac{1}{3} \cdot \frac{\ln (\frac{11}{4}) + \ln (4)}{\ln (2)}$

Step 3.3

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

$\frac{1}{3} \cdot \frac{\ln (\frac{11}{4} \cdot 4)}{\ln (2)}$

Step 3.4

Cancel the common factor of $4$ .

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Step 3.4.1

Cancel the common factor.

$\frac{1}{3} \cdot \frac{\ln (\frac{11}{4} \cdot 4)}{\ln (2)}$

Step 3.4.2

Rewrite the expression.

$\frac{1}{3} \cdot \frac{\ln (11)}{\ln (2)}$

Step 4

Multiply $\frac{1}{3} \cdot \frac{\ln (11)}{\ln (2)}$ .

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Step 4.1

Multiply $\frac{1}{3}$ by $\frac{\ln (11)}{\ln (2)}$ .

$\frac{\ln (11)}{3 \ln (2)}$

Step 4.2

Simplify $3 \ln (2)$ by moving $3$ inside the logarithm.

$\frac{\ln (11)}{\ln (2^{3})}$

Step 5

Raise $2$ to the power of $3$ .

$\frac{\ln (11)}{\ln (8)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (11)}{\ln (8)}$

Decimal Form:

$1.15314387 \dots$