Finding Inverse Exponential Functions: A Method for Solving Without Answers

[omg precal]

First of all, sorry for asking so many questions. I do not want answers, just a method of solving them.

Homework Statement

For the exponential function f(x) = ab^x, suppose f(2) = 2 and f(4) = 18.

a. Find a and b.
b. Find f^-1(54), the inverse function.

Homework Equations

None, really...

The Attempt at a Solution

a. f(2) = 2, meaning ab^2 = 2. f(4) = 18, meaning ab^4 = 18.

(ab^2 = 2) * 9 -> 9ab^2 = 18
ab^4 = 18

9ab^2 = ab^4

divide both sides by ab^2

9 = b^2
b = 3

a(3)^2 = 2
9a = 2
a = 2/9

b. The inverse of ab^54...

Inverse of: (2/9)(3)^54

And from here, I'm lost. Do you fifty-fourth root everything because it is the inverse?

 

[Doc Al]

inverse

b. Find f^-1(54), the inverse function.

This means: find the value of x such that ab^x = 54.

 

[omg precal]

that's what the inverse does to the function?

 

[omg precal]

that gives me 3^x = 243. using logarithms...

x log3 = log 243

x = log 243/ log 3

x = 5

am i correct?

 

[Doc Al]

Plug your answer into the original equation and see for yourself!

 

[omg precal]

3^5 is indeed 243.

Thanks, Doc Al.

 

[BlackWyvern]

One thing I remember about the inverse function is that it's the function that for a value of y, would return the x-value that the original function used.

f(x)
x | y
-----
1 | 3
2 | 12
3 | 27
4 | 48
5 | 75

So for
f^-1(x)
x | y
-----
3 | 1
12 | 2
27 | 3
48 | 4
75 | 5

So:

y = ab^x
Solve for x to get the inverse function:

x = lnb / (lny - lna)

I think. I'm not sure at all.