005 Sample Final A, Question 7

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

Foundations
1) What logarithm rule is relevant for dealing with the coefficients of the logarithms?
2) How do we remove the logs?
Answer:
1) One of the rules of logarithms says that $r\log(x)=\log(x^{r})$
2) The definition of logarithm tells us that if $\log _{5}(x)=y$, then $5^{y}=x$

Step 1
Use the rules of logarithms to move the 2 and the 3 to exponents. So $\log _{5}(x^{2})=\log _{5}(4^{3})$

Step 2
By the definition of logarithm, we find that $x^{2}=4^{3}$

Step 3
Taking the square root of both sides we get $x=8$

Final Answer
$x=8$

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