# How do you solve log_3z=4log_z3?

Jan 29, 2016

$z = 9$

#### Explanation:

Rewrite everything using the change of base formula.

The change of base formula provides a way of rewriting a logarithm in terms of another base, like follows:

${\log}_{a} b = {\log}_{c} \frac{b}{\log} _ c a$

In this case, the new base I will choose is $e$, so we will use the natural logarithm.

${\log}_{3} z = 4 {\log}_{z} 3$

$\implies \ln \frac{z}{\ln} 3 = \frac{4 \ln 3}{\ln} z$

Cross multiply.

$\implies {\left(\ln z\right)}^{2} = 4 {\left(\ln 3\right)}^{2}$

Take the square root of both sides.

$\implies \ln z = 2 \ln 3$

We can modify the right hand side using the rule: $b \cdot \ln a = \ln \left({a}^{b}\right)$

$\implies \ln z = \ln \left({3}^{2}\right) = \ln 9$

Use the fairly intuitive rule that if $\ln a = \ln b$, then $a = b$.

$\implies z = 9$