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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

1

1

0

a

aa

aa

aaa

xyyx

xx

yxyx

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ADDITIONAL EXPONENT LAWS

x

xx

xxx

b

a

b

a

baab

)(

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FRACTIONAL EXPONENTS

Recall that radicals can be expressed as fractional exponents. That is,

./1 nn xx

Below are some examples.

5/35/135 3

3/13

2/1

aaa

zz

bb

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LAWS OF EXPONENT WITH BASE e

If x and y are real numbers, then

1

1

*

0

e

ee

ee

eee

xyyx

xx

yxyx

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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 LOGARITHMSDefinition: 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

.01ln

lnln

lnln

lnlnln

lnlnln

1

xyx

x

yx

yxxy

y

x

yx

• 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.