Change of base from 3 to 7

Statement

Given $\log_3(x)$, explain how you can transform it into base $7$.

Solution

Use the change-of-base formula: for every base $c>0,\ c\neq1$, $\log_a(x)=\frac{\log_c(x)}{\log_c(a)}.$ Choosing $c=7$ and $a=3$ gives $\log_3(x)=\frac{\log_7(x)}{\log_7(3)}.$ In practice: divide the base-$7$ logarithm of the argument by the base-$7$ logarithm of the old base $3$. Equivalently, $\log_3(x)=\log_7(x)\cdot\log_3(7)$, since $\log_3(7)=\dfrac{1}{\log_7(3)}$.

$\boxed{\log_3(x)=\dfrac{\log_7(x)}{\log_7(3)}}$