Simplify/Condense log base 4 of 3+3 log base 4 of 2

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

Step 1

Simplify each term.

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

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

Combine $3$ and $\frac{1}{2}$ .

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

Step 2

To write $\log_{4} (3)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3

Combine fractions.

Combine $\log_{4} (3)$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

Multiply $\log_{4} (3) \cdot 2$ .

Reorder $\log_{4} (3)$ and $2$ .

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

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

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

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

$\frac{\log_{4} (9) + 3}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\log_{4} (9) + 3}{2}$

Decimal Form:

$2.29248125 \dots$