# How do you solve log_7 12 = x?

Dec 29, 2015

From $\log$ definition, we have that ${\log}_{b} a = c \iff {b}^{c} = a$

#### Explanation:

Thus,

${\log}_{7} \left(12\right) = x \iff {7}^{x} = 12$

Now, we can apply $\log$ on both sides of the equation.

$\log {7}^{x} = \log 12$

Another $\log$ rule states that ${\log}_{b} {a}^{n} = n \cdot {\log}_{b} a$, so:

$x \cdot \log 7 = \log 12$
$x = \log \frac{12}{\log} 7$ (both on base 10 or base $e$ or whatever base you end up finding convenient! It really does not matter!)

Using a calculator, as these are not exact expressions, you can find the approximate value of

$x \cong 1.277$