# Find the inverse of the function? : # h(x) = log((x+9)/(x−6)) #

##### 2 Answers

# h^(-1)(x) = 3 ( (2e^x + 3) / (e^x - 1) ) #

#### Explanation:

We have:

# h(x) = log((x+9)/(x−6)) #

To find

Writing as:

# h = log((x+9)/(x−6)) #

# :. (x+9)/(x−6) = e^h #

# :. x+9 = (x−6)e^h #

# :. x+9 = xe^h−6e^h #

# :. xe^h - x = 6e^h + 9#

# :. x(e^h - 1) = 3(2e^h + 3)#

# :. x = 3 ( (2e^h + 3) / (e^h - 1) ) #

Hence, the inverse function is:

# h^(-1)(x) = 3 ( (2e^x + 3) / (e^x - 1) ) #

I have assumed natural logarithms (base e). If base

#### Explanation:

To find the inverse, let us switch the x and y variables, denoting

Assuming

Adding a base 10 to each side of the equation to cancel out the

Multiplying both sides by

Taking all

I am aware of the other variations in which this answer could be rewritten, but you can work off of this answer to your preference.

Hope this helped!