f(x)=

13

**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