inverses and exponents

Question

f(x)=e^x has an inverse of f^-1(x)=ln^x    

Explain why e^ln(x)=ln(e^x)

 

I don't know how to explain. Thank you.

Answer 1

For most functions  , f,   that have an inverse , f^-1   we have

 

f ( f^-1(x) =  x            AND

f^-1(f(x))   = x

 

That is, if  the function f  has an inverse,  f^-1 ,      then the inverse of the inverse   f^-1  exists and is just f

 

There can domain / range issues with this, but not for  monotone increasing functions like exp(x)

Answer 2

show e^[ln(x)]=ln[e^(x)]

ln[e^(x)]=x by definition: e^x=y means ln(y)=log_e(y)=x

so e^[ln(x)]=x

take the log_e of both sides

ln{e^[ln(x)]}=ln(x)

ln(x)ln(e)=ln(x) because log_a(p^q)=qlog_a(p)

but ln(e)=1

so we have ln(x)(1)=ln(x)

                   ln(x)=ln(x)

Answer 3

exp(lnx) = x

 

ln(expx) = x

 

So exp(lnx)= ln(expx)

 

easy