Download Section 4.2 - Gordon State College

Section 4.2
Exponential and
Logarithmic Functions
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LAWS OF EXPONENTS
Laws of Exponents with General Base a: If the
base number a is positive and x and y are any real
numbers, then
a a a
x
y
a
x
a 
x y
x y
1
 x
a
a
xy
a 1
0
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ADDITIONAL EXPONENT LAWS
(ab)  a b
x
x
x
x
x
a
a
   x
b
b
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FRACTIONAL EXPONENTS
Recall that radicals can be expressed as
fractional exponents. That is,
xx .
1/ n
n
Below are some examples.
b b
1/ 2
3
zz
5
a  a
3
1/ 3
 
3 1/ 5
a
3/ 5
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LAWS OF EXPONENT WITH
BASE e
If x and y are real numbers, then
e *e  e
x
y
e
x
e 
x y
x y
1
 x
e
e
xy
e 1
0
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COMMON LOGARITHMS
Definition: The common logarithm of the
positive number x is the power to which 10 must
be raised in order to obtain the number x. It is
denoted by log10 x. Thus,
y = log10 x
means the 10y = x.
Frequently, we omit the subscript 10 and simply
write log x for the common logarithm of the
positive number x.
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NATURAL LOGARITHMS
Definition: The natural logarithm of the
positive number x is the power to which e must
be raised in order to obtain the number x. It is
occasionally denoted by loge x, but more
frequently by ln x (with l for “log” and n for
“natural”). Thus,
y = ln x means that ey = x.
NOTE: Only positive numbers have
logarithms (common or natural).
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LAWS OF LOGARITHMS
Laws of Logarithms: If x and y are positive
real numbers, then
 ln xy  ln x  ln y
 ln
x
y
 ln x  ln y
 ln 1x   ln x
 ln x  y ln x
 ln 1  0.
y
• The logarithm of a product is the
sum of the logarithms.
• The logarithm of a quotient is the
difference of the logarithms.
• The logarithm of a reciprocal is the
negative of the logarithm.
• The logarithm of a power is the
exponent times the logarithm of the
base.
• The logarithm of one is zero.
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EXPONENTS AND LOGARITHMS
AS INVERSES
Just as addition and subtraction (and
multiplication and division) undo each other,
exponentials and logarithms undo each other
also. That is,
eln x = x
and
ln ex = x.
Two functions, that undo each other are called
inverses.
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