It is important to remember that the natural logarithm function, ln(x),
and the natural exponential function, ex, 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 ex, some people find it easier to express ex, as exp(x),
so that the composition of functions is more clearly observed.
ln(exp(x)) = x     and     exp(ln(x)) = x

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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

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For help with exponential expressions on your calculator, click here.

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