Precalculus Examples

$\frac{1}{2} \cdot (\frac{\ln (\frac{10}{3})}{\ln (4)} + 1)$

Step 1

Simplify the expression.

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Step 1.1

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

$\frac{1}{2} \cdot (\frac{\ln (\frac{10}{3})}{\ln (4)} + \frac{\ln (4)}{\ln (4)})$

Step 1.2

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{\ln (\frac{10}{3}) + \ln (4)}{\ln (4)}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{1}{2} \cdot \frac{\ln (\frac{10}{3} \cdot 4)}{\ln (4)}$

Step 2.2

Multiply $\frac{10}{3} \cdot 4$ .

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Step 2.2.1

Combine $\frac{10}{3}$ and $4$ .

$\frac{1}{2} \cdot \frac{\ln (\frac{10 \cdot 4}{3})}{\ln (4)}$

Step 2.2.2

Multiply $10$ by $4$ .

$\frac{1}{2} \cdot \frac{\ln (\frac{40}{3})}{\ln (4)}$

Step 3

Multiply $\frac{1}{2} \cdot \frac{\ln (\frac{40}{3})}{\ln (4)}$ .

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Step 3.1

Multiply $\frac{1}{2}$ by $\frac{\ln (\frac{40}{3})}{\ln (4)}$ .

$\frac{\ln (\frac{40}{3})}{2 \ln (4)}$

Step 3.2

Simplify $2 \ln (4)$ by moving $2$ inside the logarithm.

$\frac{\ln (\frac{40}{3})}{\ln (4^{2})}$

Step 4

Raise $4$ to the power of $2$ .

$\frac{\ln (\frac{40}{3})}{\ln (16)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (\frac{40}{3})}{\ln (16)}$

Decimal Form:

$0.93424139 \dots$