INVERSE OF LOGARITHMIC FUNCTION

Find the inverse of logarithmic function :

Problem 1 :

y = 2 log_x3

Solution :

y = 2 log_x3

Solve for x,

y/2 = log_x3

x^(y/2) = 3

To get the value of x, we move the power to the other side of the equal sign. So, the power will become its reciprocal.

x = 3^(2/y)

Replace x = f^-1(x) , y = x

f^-1(x) = 3^(2/x)

Problem 2 :

y = log₆ 3^x 

Solution :

solving for x,

Problem 3 :

y = log₂x³

Solution :

y = log₂x³

2^y = x³

Problem 4 :

y = log₅(-2x)

Solution :

y = log₅(-2x)

Problem 5 :

y = log₆(3x)

Solution :

y = log₆(3x)

6^y = 3x

x = 6^y/3

Replace x = f^-1(x), y = x

f^-1(x) = 6^x/3

Problem 6 :

y = log₄x + 10

Solution :

y = log₄x + 10

y - 10 = log₄x

f^-1(x) = 4^x-10

Problem 7 :

y = log₂x + 6

Solution :

y = log₂x + 6

y - 6 = log₂x

Problem 8 :

y = log_x2 - 6

Solution :

y = log_x2 - 6

y + 6 = log_x2

Problem 9 :

y = 4log_x2

Solution :

y = 4log_x2

Problem 10 :

y = log₅(x+5)

Solution :

y = log₅(x+5)

5^y = x + 5

x = 5^y - 5

Put x =  f^-1(x) and y = x

 f^-1(x) = 5^x - 5

Problem 11 :

y = log(-2x)

Solution :

y = log(-2x)

Problem 12 :

y = log_1/4x⁵

Solution :

y = log_1/4x⁵

Problem 13 :

y = log_1/5x - 4

Solution :

y = log_1/5x - 4