EXPONENTIAL AND
LOGARITHMIC FUNCTIONS
Section 11-5 Common Logarithms
Definition
 Logarithms with a base 10 are called common logarithms.
If a base is not written in logarithmic form, it is
understood to be 10.
 A common logarithm is made up of two parts, the
characteristic and the mantissa
 The mantissa is the logarithm of a number between 1 and
10. Thus the mantissa is greater than 0 but less than 1.
 The characteristic is the exponent nof ten that is used to
write the number in scientific notation.
Example # 1
 Given that log 5 = .6990 evaluate each logarithm.
 1. log 50,000
2. log .005
 The mantissa is log 5.
4
 1. log 50,000 = log 5 x 10 =
(The characteristic is 4, exponent of 10)
log 5+ 4 log 10 = .6990 + 4 = 4.6990
2. log .005 = log 5 x 10 3
(The characteristic is -3)
= log 5 + -3 log 10 = .6990 + -3 = -2.3010
Example # 2
 Evaluate each expression
3
 1. y = log 3 (5) = log 3 + 3 log 5 = .47712 + 3 (.69897) = 2.57403

13
2. log
7
3
= 3 log 13 - log 7 = 3 (1.11394) - .84510 = 2.49673
Change of Base Formula
 If a, b, n are positive numbers and neither a nor b
is 1, then the following equation is true. This
comes in handy because the calculator only has
base 10 and the natural log on it!
 Loga n= logb n/logb a
 Example: Find the value of log 8 172 using the
formula.
log 8 172 = log 172 / log 8 ≈
10
10
2.23553/.90309 = 2.47542
A ntilogarithm “undoes”the log
 Sometimes the logarithm of x is known to have a value a, but x is not
known. Then x is called the antilogarithm of a, written antilog a.
 If log x = a then x = antilog a
 Example: If we know 7.84 = log x we can find x by using the antilog.

antilog 7.84 = antilog log x

69183097.09 = x
 On the calculator the antilog is 10
x
Logarithms to solve exponential equations
 We have seen that logarithms can be used to solve exponential
equations. Graphing is an alternate way of finding approximate
solutions of exponential or logarithmic equations. To do this, graph
each side of the equation as a function and find the coordinates of the
intersection of the graphs.
 Example: Solve 2
x 1
x2
=5
by graphing.
 Using our graphing calculator or setting up a table of values for each
equation, we find that the curves intersect at approximately 2.76
HW #29
Section 11-5
Pp. 730-732
#19-41 odds, 65,67