It is important to remember that the natural logarithm function, ln(x),
and the natural exponential function, e x, are inverse functions.

When a function is composed with its inverse, the starting value is returned. 
ln(ex) = x     and     eln(x) = x

When studying e x, some people find it easier to express e x, as exp(x),
so that the composition of functions is more clearly observed.
ln(exp(x)) = x     and     exp(ln(x)) = x

dividerdash

Examples:

  Simplify: Answer
1.
 ln(ex) Knowing that ln(x) and ex are inverse functions, the simplification under composition is x.
ln(ex) = ln(exp(x)) = x
2.
 ln(e) Noting that the exponent on e is 1 (the x-value is 1), and applying what we just saw in #1, we know the simplification is one.
ln(e) = ln(exp(1)) = 1
3.
 eln(x) Again, we know that ln(x) and ex are inverse functions, so the simplification under composition is x.
eln(x) = exp(ln(x)) = x
4.
 eln(7) Noting that the x-value is 7,exponent on e is 1, and applying what we just saw in #3, we know the simplification is seven.
eln(7) = exp(ln(7)) = 7
5.
 e3ln(4)

beware That "3" is interfering with the composition of the inverse functions. Move the "3" by using the log property that
ln ar = r ln a.
exm1



How to use your graphing calculator for working
with
ex and ln(x)
Click here.
ti84c
How to use
your graphing calculator for
working
with
ex and ln(x)
Click here.

divider

NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation
 and is not considered "fair use" for educators. Please read the "Terms of Use".