Math 41
Review for Exam 3
3.1 Exponential Functions

An exponential function has the form
( )
x
f x
a
=
, where a>0
and a≠1.
You need to be able to evaluate exponential functions on your calculator.
You
need to be able to graph exponential functions by hand.
All exponential functions of the
form
( )
x
f x
a
=
have a horizontal asymptote at
y=0
.
You need to be able to do a
transformation of an exponential function.
An important exponential function is
( )
x
f x
e
=
.
You need to be able to evaluate the natural exponential function on your
calculator and by hand (when possible).
One of the most useful applications of the
exponential function is interest.
You need to know the formulas for interest: 1) For
n
compoundings per year,
(1
)
nt
r
n
A
P
=
+
, where A is the future amount, P is the principal, r
is the annual interest rate (in decimal form), and t is the time in years, 2) For continuous
compounding,
rt
A
Pe
=
.
3.2
Logarithmic Functions

Remember that a logarithmic function is the inverse
function of the exponential function.
For x>0, a>0 and a≠1,
log
y
a
y
x iff x
a
=
=
.
Remember that a logarithm is an exponent
!
You need to be able to evaluate logs by
hand, and on your calculator.
Remember that
the log button on your calculator only
evaluates common logs
(base 10).
You need to be able to graph a log function by hand.
You can do this by graphing it inverse exponential function, switching the inputs and
outputs of the exponential function, and plotting the new points for the log function.
One
of the most useful log functions is the natural log function (ln
x
).
This is just the inverse
of the natural exponential function.
You need to be able to evaluate natural logs by hand
and with your calculator.
Remember that the base of the natural log function is
e
, even
though it is not explicitly written.
Properties of Exponents
:
1.
x
y
x y
a a
a
+
=
2.
x
x y
y
a
a
a

=
3.
1
x
x
a
a

=
4.
0
1
a
=
5.
(
)
x
x
x
ab
a b
=
6.
(
)
x
y
xy
a
a
=
7.
(
29
x
x
x
a
a
b
b
=
3.3 Properties of Logarithms
 You need to be able to convert an exponential expression
to a logarithmic expression, and vice versa.
You need to know the 3 change of base
formulas:
10
10
log
log
ln
log
log
log
ln
b
a
b
x
x
x
x
a
a
a
=
=
=
.
Remember that your calculator can only
evaluate logarithms base 10 (common log) or base e (natural log).
You need to know the
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View Full Documentfollowing properties of logarithms in order to expand or condense log expressions:
log(
)
log
log
ln(
)
ln
ln
log(
)
log
log
ln(
)
ln
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 Fall '07
 BRUNSDEN,VICTORW
 Math, Exponential Function, Exponential Functions

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