Write as a Single Logarithm 2 log base 4 of 3+ log base 4 of 2

$2 \log_{4} (3) + \log_{4} (2)$

Step 1

Simplify each term.

Simplify $2 \log_{4} (3)$ by moving $2$ inside the logarithm.

$\log_{4} (3^{2}) + \log_{4} (2)$

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

$\log_{4} (9) + \log_{4} (2)$

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

$\log_{4} (9) + \frac{1}{2}$

Step 2

To write $\log_{4} (9)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\log_{4} (9) \cdot \frac{2}{2} + \frac{1}{2}$

Step 3

Combine fractions.

Combine $\log_{4} (9)$ and $\frac{2}{2}$ .

$\frac{\log_{4} (9) \cdot 2}{2} + \frac{1}{2}$

Combine the numerators over the common denominator.

$\frac{\log_{4} (9) \cdot 2 + 1}{2}$

Step 4

Simplify the numerator.

Multiply $\log_{4} (9) \cdot 2$ .

Reorder $\log_{4} (9)$ and $2$ .

$\frac{2 \cdot \log_{4} (9) + 1}{2}$

Simplify $2 \cdot \log_{4} (9)$ by moving $2$ inside the logarithm.

$\frac{\log_{4} (9^{2}) + 1}{2}$

Raise $9$ to the power of $2$ .

$\frac{\log_{4} (81) + 1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\log_{4} (81) + 1}{2}$

Decimal Form:

$2.08496250 \dots$