# How do you find the exact value of log_6 root3 6?

Dec 26, 2016

${\log}_{6} {6}^{\frac{1}{3}} = \frac{1}{3}$

#### Explanation:

Let's use laws of indices first.

Another way of writing $\text{ "root3 6" }$ is $\text{ "6^(1/3)" }$

The definition of a log is:

The log of a number is the index to which the base must be raised to equal the number.
Apply this definition here: ${\log}_{6} {6}^{\frac{1}{3}}$

$\therefore {\log}_{6} {6}^{\frac{1}{3}} = \frac{1}{3}$

Or, using index from: Log form and index form are interchangeable.

${\log}_{a} b = c \Leftrightarrow {a}^{c} = b$

If ${\log}_{6} {6}^{\frac{1}{3}} = x$, then ${6}^{x} = {6}^{\frac{1}{3}}$

$x = \frac{1}{3}$