# How do you solve log_x729=6?

Dec 31, 2015

By trying random numbers, 3 is obviously correct. However, if the answer was not an integral, the root function has to be derived and used (through calculator).

#### Explanation:

We can apply the logarithmic property:

$a = {\log}_{x} \left({x}^{a}\right)$

${\log}_{x} 729 = {\log}_{x} \left({x}^{6}\right)$

Since $\log x$ is a 1-1 function for any $x > 0$ and $x \ne 1$ , the logarithms can be ruled out:

$729 = {x}^{6}$
${729}^{\frac{1}{6}} = {\left({x}^{6}\right)}^{\frac{1}{6}}$
$x = {729}^{\frac{1}{6}} = \sqrt[6]{729} = 3$