What is the inverse of #y = ln(x) + ln(x-6) # ?

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Answer 1 · 2018-06-08T18:50:04

For the inverse to be a function a domain restriction will be required:

#y'=3+-sqrt(e^x + 9)#

#y = ln(x) + ln(x-6)#

#x = ln(y) + ln(y-6)#

Apply rule: #ln(a) + ln(b) = ln(ab)#

#x = ln(y(y-6))#

#e^x = e^(ln(y(y-6)))#

#e^x = y(y-6)#

#e^x = y^2-6y#

complete the square:

#e^x + 9 = y^2-6y +9#

#e^x + 9 = (y-3)^2#

#y-3=+-sqrt(e^x + 9)#

#y=3+-sqrt(e^x + 9)#