Evaluate 1/3* log base 4 of 8+ log base 4 of 15

$\frac{1}{3} \cdot \log_{4} (8) + \log_{4} (15)$

Step 1

Simplify each term.

Logarithm base $4$ of $8$ is $\frac{3}{2}$ .

$\frac{1}{3} \cdot \frac{3}{2} + \log_{4} (15)$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{1}{3} \cdot \frac{3}{2} + \log_{4} (15)$

Rewrite the expression.

$\frac{1}{2} + \log_{4} (15)$

Step 2

To write $\log_{4} (15)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} + \log_{4} (15) \cdot \frac{2}{2}$

Step 3

Combine fractions.

Combine $\log_{4} (15)$ and $\frac{2}{2}$ .

$\frac{1}{2} + \frac{\log_{4} (15) \cdot 2}{2}$

Combine the numerators over the common denominator.

$\frac{1 + \log_{4} (15) \cdot 2}{2}$

Step 4

Simplify the numerator.

Multiply $\log_{4} (15) \cdot 2$ .

Reorder $\log_{4} (15)$ and $2$ .

$\frac{1 + 2 \cdot \log_{4} (15)}{2}$

Simplify $2 \cdot \log_{4} (15)$ by moving $2$ inside the logarithm.

$\frac{1 + \log_{4} (15^{2})}{2}$

Raise $15$ to the power of $2$ .

$\frac{1 + \log_{4} (225)}{2}$

Step 5

Simplify the numerator.

Rewrite $\log_{4} (225)$ using the change of base formula.

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{1 + \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}{2}$

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{1 + \frac{\log (225)}{\log (4)}}{2}$

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

$\frac{\frac{\log (4)}{\log (4)} + \frac{\log (225)}{\log (4)}}{2}$

Combine the numerators over the common denominator.

$\frac{\frac{\log (4) + \log (225)}{\log (4)}}{2}$

Simplify the numerator.

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

$\frac{\frac{\log (4 \cdot 225)}{\log (4)}}{2}$

Multiply $4$ by $225$ .

$\frac{\frac{\log (900)}{\log (4)}}{2}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log (900)}{\log (4)} \cdot \frac{1}{2}$

Step 7

Multiply $\frac{\log (900)}{\log (4)} \cdot \frac{1}{2}$ .

Multiply $\frac{\log (900)}{\log (4)}$ by $\frac{1}{2}$ .

$\frac{\log (900)}{\log (4) \cdot 2}$

Reorder $\log (4)$ and $2$ .

$\frac{\log (900)}{2 \cdot \log (4)}$

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

$\frac{\log (900)}{\log (4^{2})}$

Step 8

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

$\frac{\log (900)}{\log (16)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (900)}{\log (16)}$

Decimal Form:

$2.45344529 \dots$

Evaluate 1/3* log base 4 of 8+ log base 4 of 15

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