Simplify/Condense 3 log base 4 of 2-5 log base 4 of 11

$3 \log_{4} (2) - 5 \log_{4} (11)$

Step 1

Simplify each term.

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

$3 (\frac{1}{2}) - 5 \log_{4} (11)$

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

$\frac{3}{2} - 5 \log_{4} (11)$

Simplify $- 5 \log_{4} (11)$ by moving $5$ inside the logarithm.

$\frac{3}{2} - \log_{4} (11^{5})$

Raise $11$ to the power of $5$ .

$\frac{3}{2} - \log_{4} (161051)$

Step 2

To write $- \log_{4} (161051)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{2} - \log_{4} (161051) \cdot \frac{2}{2}$

Step 3

Combine $- \log_{4} (161051)$ and $\frac{2}{2}$ .

$\frac{3}{2} + \frac{- \log_{4} (161051) \cdot 2}{2}$

Step 4

Simplify the expression.

Combine the numerators over the common denominator.

$\frac{3 - \log_{4} (161051) \cdot 2}{2}$

Multiply $2$ by $- 1$ .

$\frac{3 - 2 \log_{4} (161051)}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{3 - 2 \log_{4} (161051)}{2}$

Decimal Form:

$- 7.14857904 \dots$