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

Examples:

 Simplify: Answer 1. ln(ex) Knowing that ln(x) and ex are inverse functions, the simplification under composition is x. 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(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(7) 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.