Calculus of a Single Variable, Chapter 5, 5.5, Section 5.5, Problem 34

To simplify the logarithmic equation: log_5(sqrt(x-4))=3.2 , recall the logarithm property: a^((log_(a)(x))) = x .
When a logarithm function is raised by the same base, the log cancels out which is what we need to do on the left side of the equation.
As a rule we apply same change on both sides of the equation.
Raising both sides by base of 5:
5^(log_5(sqrt(x-4)))= 5^(3.2)
sqrt(x-4) = 5^(3.2)
To cancel out the radical sign, square both sides:
(sqrt(x-4))^2 = (5^(3.2)) ^2
x-4 =5^(6.4)
x= 5^(6.4)+4
x~~29748.593 (rounded off to three decimal places)
To check, plug-in x=29748.593 in log_5(sqrt(x-4)) :
log_5(sqrt(29748.593-4))
log_5(sqrt(29744.593))
log_5(172.4662083)=3.2 which is what we want

So, x=29748.593 is the real solution.
Note: (x^m)^n= x^((m*n ))