How do you find the inverse of #y = ln(x) + ln(x-6)#?

Question

Lily Anderson · Accepted Answer

Solve for #x# using properties of logarithms and the quadratic formula and eliminate an extraneous solution to find
#f^-1(x) = 3 + sqrt(36-4e^x)/2# We'll proceed under the assumption you are trying to find the inverse of the function #f(x) = ln(x) + ln(x-6)# In general, to find the inverse of a function, a good method is to set #y = f(x)# and then solve for #x# to obtain #x = f^(-1)(y)# (To see why this works, substitute in #f(x)# for #y# and note that the result is #f^(-1)(f(x)) = x# as desired.) To do that here, we will need to use the following: Let #y = f(x) = ln(x) + ln(x-6)# (note here that as we have #ln(x-6)# it must be that #x > 6#) #=> y = ln(x(x-6)) = ln(x^2 - 6x)# #=> e^y = e^(ln(x^2 - 6x)) = x^2 - 6x# #=> x^2 - 6x - e^y = 0# #=> x = (-(-6)+-sqrt((-6)^2-4(1)(-e^y)))/(2(1))# #= (6 +-sqrt(36 + 4e^y))/2# #= 3 +-sqrt(36 + 4e^y)/2# But #3 - sqrt(36+4e^y)/2 < 6#, so, as noted above, we must throw it out as a possible solution for #x# Thus #x = 3 + sqrt(36-4e^y)/2# Then, by our process, we have #f^-1(x) = 3 + sqrt(36-4e^x)/2#

Cora Alexander · Answer

undefined To find the inverse of the function $ y = \ln(x) + \ln(x - 6) $, we switch the roles of $ x $ and $ y $ and then solve for $ y $.

Step 1: Swap $ x $ and $ y $ to get $ x = \ln(y) + \ln(y - 6) $.

Step 2: Combine the logarithmic terms using the properties of logarithms.

$$ x = \ln(y(y - 6)) $$

Step 3: Exponentiate both sides to eliminate the natural logarithm.

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

Step 4: Expand and rearrange the equation.

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

Step 5: Rearrange the equation into standard quadratic form (ax^2 + bx + c = 0).

$$ y^2 - 6y - e^x = 0 $$

Step 6: Use the quadratic formula to solve for $ y $:

$$ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

For our equation, $ a = 1 $, $ b = -6 $, and $ c = -e^x $.

$$ y = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(-e^x)}}{2(1)} $$

$$ y = \frac{6 \pm \sqrt{36 + 4e^x}}{2} $$

Step 7: Simplify the expression:

$$ y = \frac{6 \pm \sqrt{36 + 4e^x}}{2} $$

Therefore, the inverse of the function $ y = \ln(x) + \ln(x - 6) $ is given by:

$$ y = \frac{6 \pm \sqrt{36 + 4e^x}}{2} $$