If two logarithms are equal to each other, then their arguments are equal if and only if their bases are equal.
See solution.
Practice makes perfect
First we will describe the error made, and then we will solve the equation in a correct way.
Describing the Error
Let's analyze the given solution step by step, and look for the error made.
log_2x=2log_39 ⇔ log_2x=log_39^2
The first step is correct. The Power Property of Logarithms was used to rewrite the right-hand side of the equation. Let's take a look at the next step.
log_2x=log_39^2 ⇔ x=9^2
This is the step in which the error was made. Recall that if two logarithms are equal to each other, then their arguments are equal if and only if their bases are equal. In this case the base of the first logarithm is 2 and the base of the other one is 3. Therefore, we cannot claim that their arguments are equal.
Correcting the Error
Now we will solve the given equation.
log_2x=2log_39
In this case, we are able to evaluate the logarithm on the right-hand side of the equation. Let's do it!
Finally, we will rewrite the obtained equation using the definition of a logarithm.
log_2x=4 ⇔ x= 2^4
Therefore, x=2^4=16 is the correct solution to the equation.
