# How do you evaluate \log _ { 3} 2+ 3\log _ { 3} 4?

Aug 6, 2017

$4.4165$

#### Explanation:

If you are adding logs with the same base, it it the same as multiplying the numbers.

${\log}_{3} 2 + 3 {\log}_{3} 4 \text{ } \leftarrow$ apply the power law.

$= {\log}_{3} 2 + {\log}_{3} {4}^{3}$

$= {\log}_{3} \left(2 \times 64\right)$

${\log}_{3} 128$

$128$ is not a power of $3$, so to continue requires a calculator.
${3}^{4} = 81 \mathmr{and} {3}^{5} = 243$, so the value has to be between $4 \mathmr{and} 5$
$\frac{{\log}_{10} 128}{{\log}_{10} 3}$
$= 4.4165$