HOW TO FIND INVERSE OF A LOGARITHMIC FUNCTION

Before learning how to find inverse of a logarithmic function, you need to know how to convert an equation from logarithmic to exponential.

Consider the following equation in logarithmic form.

log_am = x

The picture below illustrates how to convert the above equation from logarithmic to exponential form.

Example 1 :

Find f^-1(x), if f(x) = log₂(x). 

Solution :

f(x) = log₂(x)

Replace f(x) by y.

y = log₂(x)

Interchange x and y. 

x = log₂(y)

The above equation is in logarithmic form. Convert it to exponential form.

2^x = y

or

y = 2^x

Replace y by f^-1(x).

f^-1(x) = 2^x

Example 2 :

Find f^-1(x), if f(x) = log₅(5x + 2). 

Solution :

f(x) = log₅(5x + 2)

Replace f(x) by y.

y = log₅(5x + 2)

Interchange x and y. 

x = log₅(5y + 2)

The above equation is in logarithmic form. Convert it to exponential form.

5^x = 5y + 2

or

5y + 2 = 5^x

Subtract 2 from both sides.

5y = 5^x - 2

Divide both sides by 5.

y = (5^x - 2)/5

Replace y by f^-1(x).

f^-1(x) = (5^x - 2)/5

Example 3 :

Find g^-1(x), if g(x) = -2ln(5 - 2x) + 8. 

Solution :

g(x) = -2ln(5 - 2x) + 8

Replace g(x) by y.

y = -2ln(5 - 2x) + 8

Interchange x and y. 

x = -2ln(5 - 2y) + 8

Subtract 8 from both sides.

x - 8 = -2ln(5 - 2y)

Divide both sides by -2.

(x - 8)/(-2) = ln(5 - 2y)

(8 - x)/2 = ln(5 - 2y)

In the equation above, we have natural logarithm 'ln' and its base is e. Convert it to exponential form.

e^{(8 - x)/2} = 5 - 2y

Add 2y to both sides.

e^{(8 - x)/2} + 2y = 5

Subtract e^{(8 - x)/2} from both sides.

2y = 5 - e^{(8 - x)/2}

Divide both sides by 5.

y = (5 - e^{(8 - x)/2})/2

Replace y by g^-1(x).

g^-1(x) = (5 - e^{(8 - x)/2})/2

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