# How do you evaluate log_3 7 using the change of base formula?

Jan 7, 2017

I found: $1.77124$
The change of base allows you to change from a base, say $b$, to a new base $c$ as:
${\log}_{b} \left(x\right) = \frac{{\log}_{c} \left(x\right)}{{\log}_{c} \left(b\right)}$
Where the new base $c$ can be choosen to be "easy" to evaluate; if you have a pocket calculator, the new base could be $e$ that can be evaluated using the Natural Logarithm ($\ln$) on the calculator.
${\log}_{3} \left(7\right) = \frac{{\log}_{e} \left(7\right)}{{\log}_{e} \left(3\right)} = \frac{\ln \left(7\right)}{\ln \left(3\right)} = 1.77124$