Exponential Expressions
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An exponential expression is one which contains an exponent. 

Since we have already dealt with many exponential expressions in the unit on "Negative and Fractional Exponents", this section will be devoted to working with exponential expressions that contain e.

e is an irrational number, approximately 2.71828183, named after the 18th century Swiss mathematician, Leonhard Euler.

ex, such as  f (x) = ex,  is called the natural exponential function.

It is important to remember that the natural logarithm function, ln, and the natural exponential function, ex, are inverse functions.  (For more information on these functions see Exponential and Logarithmic Functions.)  When a function is composed with its inverse, the starting value is returned. 

Examples:

 

  Simplify:

  Answer

1.    Since ln and ex are inverse functions, the answer, under this composition, is x.
ln ex  = ln(exp(x)) = x
2.    Since the exponent on e is 1, the answer is one.
ln e  = ln(exp(1)) = 1
3.    Since ln and ex are inverse functions, the answer, under this composition, is x.
eln x  = exp(ln x) = x
4.    eln 4  = exp(ln 4) = 4
5.    Be careful on this one!!!  That "2" is interfering with the composition of the inverse functions.  Move the "2" by using the property ln ar = r ln a.


Hint:  When first studying ex, some people find it easier to express
ex as exp(x)
so that the composition of functions is more clearly observed.

 


 

How to use your
TI-83+/84+ graphing calculator with exponential expressions.
Click calculator.