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:

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

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

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

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

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