# How do you solve log_3 2 + log_3 7 = log_3 x?

Jul 23, 2016

x = 14

#### Explanation:

Using the $\textcolor{b l u e}{\text{laws of logarithms}}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\log x + \log y = \log \left(x y\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}} \ldots \ldots . . \left(1\right)$
This law applies to logarithms to any base.

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\log x = \log y \Rightarrow x = y} \textcolor{w h i t e}{\frac{a}{a}} |}}} \ldots \ldots . . \left(2\right)$
This law applies to logarithms with equal bases.

Using (1) on left side

${\log}_{3} 2 + {\log}_{3} 7 = {\log}_{3} \left(2 \times 7\right) = {\log}_{3} 14$

Using (2)

${\log}_{3} 14 = {\log}_{3} x \Rightarrow x = 14$