Page 163 - How To Solve Word Problems In Calculus

Chapter 6

Exponential Functions

x
If b is positive and x is real, the function f (x) = b is called
an exponential function. If the base of the exponential is
e ≈ 2.71828, the function is known as the natural exponential
function. Its inverse is the natural logarithm function, usually
represented as ln x.
Listed below are some of the basic properties of the nat-
ural exponential and logarithm functions.

x
y = e if and only if x = ln y
0
e = 1 ln 1 = 0
1
e = e ln e = 1
x y
e x+y = e e ln xy = ln x + ln y
e x x
e x−y = ln = ln x − ln y
e y y
x y
y
(e ) = e xy ln x = y ln x
It is also useful to remember that as direct consequences of
1 x y
the above properties, ln =− lnx and − ln = ln . Because
x y x
x
x
e and ln x are inverse functions e ln x = x and ln(e ) = x.
The graphs of the exponential functions e x and e −x
are shown below for reference. These functions play a

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