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How do you find the inverse of #y=e^x#?

2 Answers
Jan 15, 2016

Answer:

The inverse function is #lnx#.

Explanation:

By definition, #y=f^(-1)(x)ifff(y)=x#

#iffe^y=x#

#iffy=lnx#.

Jan 15, 2016

Answer:

#x=ln(y)#

Explanation:

Given
#color(white)("XXX")y=e^x#

Taking the natural logarithm of both sides
#color(white)("XXX")color(red)(ln(y)=ln(e^x))#

By definition of #ln#
#color(white)("XXX")ln(a)= # the value. #b#, needed to make #e^b=a#
(you should memorize this)
therefore
#color(white)("XXX")ln(e^x)= # the value,# b#, needed to make #e^b=e^x#
that is
#color(white)("XXX")ln(e^x)=x#

Therefore #color(red)(ln(y)=ln(e^x))#
implies
#color(white)("XXX")color(blue)(x=ln(y))#