# 005 Sample Final A, Question 7

Question Solve the following equation,      ${\displaystyle 2\log _{5}(x)=3\log _{5}(4)}$

Foundations
1) What logarithm rule is relevant for dealing with the coefficients of the logarithms?
2) How do we remove the logs?
1) One of the rules of logarithms says that ${\displaystyle r\log(x)=\log(x^{r})}$
2) The definition of logarithm tells us that if ${\displaystyle \log _{5}(x)=y}$, then ${\displaystyle 5^{y}=x}$
Use the rules of logarithms to move the 2 and the 3 to exponents. So ${\displaystyle \log _{5}(x^{2})=\log _{5}(4^{3})}$
By the definition of logarithm, we find that ${\displaystyle x^{2}=4^{3}}$
Taking the square root of both sides we get ${\displaystyle x=8}$
${\displaystyle x=8}$