How do you find the inverse of #f(x)=ln(2+ln(x))#?

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Answer 1 · 2016-07-18T17:16:35

#x = e^{e^y-2}#

Calling

#y = log_e(2+log_e(x))#

we have also

#e^y = 2+log_e(x)#

and #log_e(x e^2) = e^y# then

#xe^2=e^{e^y}# and finally

#x = e^{e^y-2}#

So with this procedure we obtained a function #g# such that

#x = g(y)#.

Now we can operate

#y = f(x)=f(g(y))=f@g(y)# such that

#f@g equiv 1#. Here #g# is called inverse function regarding #f#

Answer 2 · 2016-07-18T17:21:49

Inverse function of #f(x)=y=ln(2+lnx)# is
#f(x)=e^(e^x-2)#

Let #f(x)=y=ln(2+lnx)#.

Hence, #2+lnx=e^y# or

#lnx=e^y-2# and

#x=e^(e^y-2)#

Hence inverse function of #f(x)=ln(2+lnx)# is
#f(x)=e^(e^x-2)#