FIND INVERSE OF LOGARITHMIC FUNCTION

If ƒ is a one-to-one function with domain D and range R, then the inverse function of ƒ, denoted by f^-1, is the function with domain R and range D defined by

ƒ^-1(b) = a if and only if ƒ(a) = b

To find inverse of a logarithmic function, we follow the steps given below.

Step 1 :

Replace f(x) by y.

Step 2 :

Derive the function for x.

Step 3 :

Replace x by f^-1(x) and y by x.

Find the inverse of each of the following functions.

Example 1 :

f(x) = log₂(x-3) - 5

Solution :

Example 2 :

f(x) = 3log₃(x+3) + 1

Solution :

Example 3 :

f(x) = -2 log 2(x - 1) + 2

Solution :

Example 4 :

f(x) = -ln(1 - 2x) + 1

Solution :

Example 5 :

f(x) = 2^x - 3

Solution :

Let y = f(x)

y = 2^x - 3

Add 3 on both sides

y + 3 = 2^x

log₂(y + 3) = x

f^-1(x) = log₂(x + 3)

Example 6 :

f(x) = 2 ⋅3^3x - 1

Solution :

Example 7 :

f(x) = -5 ⋅e^x + 2

Solution :

y = -5 ⋅e^x + 2

y - 2 = -5 ⋅e^x

2 - y = 5e^x

(2 - y)/5 = e^x

x = ln [(2 - y)/5]

f^-1(x) = ln [(2 - x)/5]

Example 8 :

f(x) = 1 - 2 ⋅e^-2x

Solution :

y = 1 - 2 ⋅e^-2x

2 ⋅e^-2x = 1 - y

e^-2x = (1 - y)/2

-2x = ln [(1 - y)/2]

x = (-1/2) ln [(1 - y)/2]

f^-1(x) = (-1/2) ln [(1 - x)/2]