Calculus of a Single Variable, Chapter 5, 5.5, Section 5.5, Problem 33

To solve a logarithmic equation, we may simplify or rewrite it using the properties of logarithm.
For the given problem log_(3)(x^2)=4.5 , we may apply the property:
a^((log_(a)(x))) = x
The "log" cancels out which we need to accomplish on the left side of the equation.
Raising both sides by the base of 3:
3^((log_3(x^2))) = 3^(4.5)
x^2= 3^(4.5)
Taking the square root on both sides:
sqrt(x^2) =+-sqrt(3^(4.5))
x= +-11.84466612
Rounded off to three decimal places:
x=+-11.845 .
Plug-in the x-values to check if they are the real solution:

log_3(11.845^2)=4.5 so x = 11.845 is a real solution.

Now let x=-11.845
log_3((-11.845)^2)
log_3(140.304025)=4.5 so x = -11.845 is a real solution.

So, x= 11.845, x = -11.845 are both solutions.