Exponential and Logrithmic Functions

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EXPONENTIAL AND LOGARITHMIC FUNCTIONS
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EXPONENTIAL FUNCTION If x and b are real numbers
such that b gt 0 and b ? 1, then f(x) bx is an
exponential function with base b.
Examples of exponential functions a) y 3x
b) f(x) 6x c) y
2x
Example Evaluate the function y 4x at the
given values of x. a) x 2
b) x -3 c) x 0
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• PROPERTIES OF EXPONENTIAL FUNCTION y bx
• The domain is the set of all real numbers.
• The range is the set of positive real numbers.
• The y intercept of the graph is 1.
• The x axis is an asymptote of the graph.
• The function is one to one.

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The graph of the function y bx
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EXAMPLE 1 Graph the function y 3x
X -3 -2 -1 0 1 2 3
y 1/27 1/9 1/3 1 3 9 27
1
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EXAMPLE 2 Graph the function y (1/3)x
X -3 -2 -1 0 1 2 3
y 27 9 3 1 1/3 1/9 1/27
1
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NATURAL EXPONENTIAL FUNCTION f(x) ex
1
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LOGARITHMIC FUNCTION For all positive real
numbers x and b, b ? 1, the inverse of the
exponential function y bx is the logarithmic
function y logb x. In symbol, y logb x
if and only if x by
Examples of logarithmic functions a) y log3
x b) f(x) log6 x c) y log2 x
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EXAMPLE 1 Express in exponential form
EXAMPLE 2 Express in logarithmic form
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• PROPERTIES OF LOGARITHMIC FUNCTIONS
• The domain is the set of positive real
  numbers.
• The range is the set of all real numbers.
• The x intercept of the graph is 1.
• The y axis is an asymptote of the graph.
• The function is one to one.

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The graph of the function y logb x
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EXAMPLE 1 Graph the function y log3 x
X 1/27 1/9 1/3 1 3 9 27
y -3 -2 -1 0 1 2 3
1
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EXAMPLE 2 Graph the function y log1/3 x
X 27 9 3 1 1/3 1/9 1/27
y -3 -2 -1 0 1 2 3
1
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• PROPERTIES OF EXPONENTS
• If a and b are positive real numbers, and m and n
  are rational numbers, then the following
  properties holds true

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• To solve exponential equations, the following
  property can be used
• bm bn if and only if m n and b gt 0,
  b ? 1

EXAMPLE 1 Simplify the following
EXAMPLE 2 Solve for x
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PROPERTIES OF LOGARITHMS If M, N, and b (b ? 1)
are positive real numbers, and r is any real
number, then
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Since logarithmic function is continuous and
one-to-one, every positive real number has a
unique logarithm to the base b. Therefore,
logbN logbM if and only if N
M
EXAMPLE 1 Express the ff. in expanded form
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EXAMPLE 2 Express as a single logarithm
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NATURAL LOGARITHM Natural logarithms are to the
base e, while common logarithms are to the base
10. The symbol ln x is used for natural
logarithms.
EXAMPLE Solve for x
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CHANGE-OF-BASE FORMULA
EXAMPLE Use common logarithms and natural
logarithms to find each logarithm
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Solving Exponential Equations Guidelines 1.
Isolate the exponential expression on one side of
the equation. 2. Take the logarithm of each
side, then use the law of logarithm to bring down
the exponent. 3. Solve for the variable.
EXAMPLE Solve for x
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Solving Logarithmic Equations Guidelines 1.
Isolate the logarithmic term on one side of the
equation you may first need to combine the
logarithmic terms. 2. Write the equation in
exponential form. 3. Solve for the variable.
EXAMPLE 1 Solve the following
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EXAMPLE Solve for x
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• Application (Exponential and Logarithmic
  Equations)
• The growth rate for a particular bacterial
  culture can be calculated using the formula B
  900(2)t/50, where B is the number of bacteria and
  t is the elapsed time in hours. How many bacteria
  will be present after 5 hours?
• How many hours will it take for there to be
  18,000 bacteria present in the culture in example
  (1)?
• A fossil that originally contained 100 mg of
  carbon-14 now contains 75 mg of the isotope.
  Determine the approximate age of the fossil, to
  the nearest 100 years, if the half-life of
  carbon-14 is 5,570 years.

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• In a town of 15,000 people, the spread of a rumor
  that the local transit company would go on strike
  was such that t hours after the rumor started,
  f(t) persons heard the rumor, where experience
  over time has shown that
• a) How many people started the rumor?
• b) How many people heard the rumor after 5
  hours?
• 5. A sum of 5,000 is invested at an interest
  rate of 5 per year. Find the time required for
  the money to double if the interest is compounded
  (a) semi-annually (b) continuously.