f^(-1)(x)=50(1.15)^x,find f(x)?

The inverse function of an inverse function is the original function. So let's get the inverse of this:

f⁻¹(x) = 50(1.15)^x

Invert the variables and change the inverse function into the function:

x = 50(1.15)^f(x)

Now solve for f(x), starting with dividing both sides by 50:

x / 50 = 1.15^f(x)

Now we can get the log of both sides:

ln(x / 50) = ln(1.15^f(x))

The exponent can now be pulled out:

ln(x / 50) = f(x) ln(1.15)

And one last division:

f(x) = ln(x / 50) / ln(1.15)

Of course, this is only true if x > 0.

As a test, let's give x a value and find f⁻¹(x). Let's use x = 2:

f⁻¹(x) = 50(1.15)^x
f⁻¹(2) = 50(1.15)^2
f⁻¹(2) = 50(1.3225)
f⁻¹(2) = 66.125

Now if we put that into our function, we should get 2 back:

f(x) = ln(x / 50) / ln(1.15)
f(66.125) = ln(66.125 / 50) / ln(1.15)
f(66.125) = ln(1.3225) / ln(1.15)

Using decimal approximations and rounding here, but in my calculator, I'm keeping it as-is to reduce errors due to rounding:

f(66.125) = 0.27952 / 0.13976
f(66.125) = 2

And we are back to where we started, so this is the correct function:

f(x) = ln(x / 50) / ln(1.15)

y = 50 * 1.15^x
The inverse is:
x = 50 * 1.15^y
0.02x = 1.15^y
Assuming all variables are real numbers:
y = log[1.15](0.02x)

F^(-1)(x)=50(1.15)^x , y = f(x)
x=50(1.15)^y , solve for y

Switch x with y and solve for y

y = 50 * 1.15^x
x = 50 * 1.15^y
x/50 = 1.15^y
ln(x/50) = y * ln(1.15)
y = ln(x/50) / ln(1.15)
y = (ln(x) - ln(50)) / (ln(23) - ln(20))
y = (ln(x) - ln(2) - 2 * ln(5)) / (ln(23) - 2 * ln(2) - ln(5))