Find F^1(x) if f(x)=ln(ln2x)?

1 Answer
Feb 23, 2018

#f^-1(x)=1/2e^(e^x)#

Explanation:

F^1 = #f^-1(x)#

I am assuming this is supposed to be the inverse.

#f(x)=ln(ln(2x)#

To find the inverse we need to make #y# a function of #x#:

#y=ln(ln(2x)#

Make both sides a power of #bbe#:

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

#e^y=ln(2x)#

Reason:

#color(red)("If" color(white)(88)y=lna)#

#color(red)("Then:")#

#color(red)(e^y=a)#

#color(red)(e^(lna)=a)#

#color(red)("So:")#

#color(red)(e^(ln(ln(2x))=ln(2x))#

#e^y=ln(2x)#

Making both sides a power of #bbe# again:

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

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

Divide by #2#:

#1/2e^(e^y)=x#

Substitute #y=x#:

#y=1/2e^(e^x)#

#f^-1(x)=1/2e^(e^x)#