Find the inverse of y = ln (x/(x-1))
To find the inverse of the given function, y = ln (x / (x-1)), let's first switch the roles of x and y:
x = ln (y / (y-1))
Step 1: Solve for y
To isolate the logarithm, let's exponentiate both sides of the equation with base e:
e^x = y / (y-1)
Step 2: Solve for y
Multiply both sides of the equation by (y-1):
e^x * (y-1) = y
Expand the left side:
e^x * y - e^x = y
Rearrange the terms to isolate y:
e^x * y - y = e^x
Factor out y:
y * (e^x - 1) = e^x
Divide both sides by (e^x - 1):
y = e^x / (e^x - 1)
Therefore, the inverse of the given function is:
g(x) = e^x / (e^x - 1)