Answer:
Okay, let's find the equation of the inverse of the function f(x) = 2 log₅(x-3) + 4.
Given:
f(x) = 2 log₅(x-3) + 4
To find the inverse function f^(-1)(x), we can follow the steps:
Step 1: Start with the given function f(x).
f(x) = 2 log₅(x-3) + 4
Step 2: Replace f(x) with y.
y = 2 log₅(x-3) + 4
Step 3: Swap x and y.
x = 2 log₅(y-3) + 4
Step 4: Solve the equation for y.
x - 4 = 2 log₅(y-3)
(x - 4)/2 = log₅(y-3)
5^((x - 4)/2) = y - 3
Step 5: Solve for y in terms of x.
y = 5^((x - 4)/2) + 3
Therefore, the equation of the inverse function f^(-1)(x) is:
f^(-1)(x) = 5^((x - 4)/2) + 3
Now, to find the equation when f(x) = 13, we can substitute 13 for f(x) in the original function:
13 = 2 log₅(x-3) + 4
9 = 2 log₅(x-3)
log₅(x-3) = 9/2
x - 3 = 5^(9/2)
x = 5^(9/2) + 3
Therefore, the equation of the inverse function f^(-1)(13) is:
f^(-1)(13) = 5^((5^(9/2) + 3 - 4)/2) + 3
f^(-1)(13) = 5^((5^(9/2) - 1)/2) + 3