Simplify/Condense 2( log base 3 of 20- log base 3 of 4)+0.5 log base 3 of 4

$2 (\log_{3} (20) - \log_{3} (4)) + 0.5 \log_{3} (4)$

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$2 \log_{3} (\frac{20}{4}) + 0.5 \log_{3} (4)$

Divide $20$ by $4$ .

$2 \log_{3} (5) + 0.5 \log_{3} (4)$

Simplify $2 \log_{3} (5)$ by moving $2$ inside the logarithm.

$\log_{3} (5^{2}) + 0.5 \log_{3} (4)$

Raise $5$ to the power of $2$ .

$\log_{3} (25) + 0.5 \log_{3} (4)$

Simplify $0.5 \log_{3} (4)$ by moving $0.5$ inside the logarithm.

$\log_{3} (25) + \log_{3} (4^{0.5})$

Raise $4$ to the power of $0.5$ .

$\log_{3} (25) + \log_{3} (2)$

Log base $3$ of $2$ is approximately $0.63092975$ .

$\log_{3} (25) + 0.63092975$

The result can be shown in multiple forms.

Exact Form:

$\log_{3} (25) + 0.63092975$

Decimal Form:

$3.56087679 \dots$