RATIONAL FUNCTIONS

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PRECALCULUS I
LOGARITHMIC FUNCTIONS
Dr. Claude S. MooreDanville Community College
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DEFINITION
The logarithmic function isf(x) loga x where
y loga x iff x a y with any real number y,
x gt 0, and 0 lt a ¹ 1.
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SPECIAL PROPERTIES
1. log a 1 0 because a0 1. 2. log a a 1
because a1 a. 3. log a a x x because a x a
x 4. If log a x log a y, then x y
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NATURAL LOGARITHMIC FUNCTION
The natural logarithmic function is f(x) loge x
ln x where x gt 0.
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SPECIAL PROPERTIES OF NATURAL LOG, ln
1. ln 1 0 because e 0 1. 2. ln e 1 because
e 1 e. 3. ln e x x because e x e x 4. If ln
x ln y, then x y
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EXAMPLE MORTGAGE PAYMENT
The length, t years, of a 150,000-mortgage at
10 with monthly payments of x (gt1250) is
approximated by
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FORMULA DERIVATION
Solving for t yields
P monthly payment A amount of loan r rate
(), n of payments/yr t years
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FORMULA DERIVATION
Solving for t yields the formula below
P monthly payment A amount of loan r rate
(), n of payments/yr t years
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EXAMPLE continued MORTGAGE PAYMENT
How many years are needed with monthly payments
of 1500?
Answer is
t 17.99 or 18 years. (Total paid 323,871.27.)
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