Finding Inverse of natural logarithm Function

Question

Find the inverse of F(x) = 2+3ln(x) / 4-lnx

Answer 1

F(x) = (2+3ln(x)) /(4-lnx).

Domain of this function is: x > 0 and lnx ≠ 4, x ≠ e^4, so domain is (0, e⁴)U(e⁴, ∞).

Now y = ( 2+3ln(x)) / (4-lnx); then interchange x and y: x = (2+3ln(y) /(4-lny),

lny = ((4x - 2)/(x + 3) any = F^-1(x) = e^{(4x - 2)/(x + 3)}, x ≠ - 3

Answer 2

F(x) = y = (2 + 3lnx) / (4 - lnx)

To find the inverse, switch x and y and solve for y:

x = (2 + 3lny) / (4 - lny)

x (4 - lny) = 2 + 3lny

4x - xlny = 2 + 3lny

4x - 2 = xlny + 3lny

4x - 2 = (x + 3)lny

lny = (4x - 2) / (x + 3)

e^lny = e^{(4x - 2) / (x + 3)}

y = e^{(4x - 2) / (x + 3)}

F^-1(x) = e^{(4x - 2) / (x + 3)}