Transcript Angles, Degrees, and Special Triangles

Logarithmic Functions and Their
Graphs
MATH 109 - Precalculus
S. Rook
Overview
• Section 3.2 in the textbook:
– Logarithmic functions
– Basic logarithmic properties
– Graphing logarithmic functions
– Domain of logarithmic functions
– Common and natural logartihms
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Logarithmic Functions
Logarithmic Functions
• Because the graph of f(x) = ax passes the horizontal
line test, it has an inverse
• If we apply the procedure for obtaining an inverse,
we get x = ay
– No way to solve this explicitly for y so we need a new
notation to express the inverse of f(x) = ax
• y = logax if and only if ay = x
y = logax is read as the logarithm base a of x is y
• f(x) = logax is called a logarithmic function and like
exponential functions has many applications
– Ex: Richter scale, decibel scale, pH level
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Exponential & Logarithmic Forms
• Two ways to express the same thing
– Sometimes one way is simpler to manipulate than
the other
• Exponential form: y = logax ↔ ay = x
• Logarithmic form: ay = x ↔ y = logax
• Need to feel comfortable converting back and
forth between exponential and logarithmic
forms
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Exponential & Logarithmic Forms
(Example)
Ex 1: Write the logarithmic equations in
exponential form and the logarithmic
equations in exponential form:
a) log4 64  3
1
 2
b) log 7
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c) 53  125
d) 811 4  3
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Evaluating Logarithms (Example)
Ex 2: Evaluate the logarithm without a
calculator:
a) log636
b) log4(1/64)
c) log3/4(9⁄16)
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Exponential and Logarithmic Functions
as Inverses
• Let f(x) = logax and g(x) = ax where a > 0, a ≠ 1,
x > 0. Then:
(f ◦ g)(x) = loga(ax) = x
(g ◦ f)(x) = a(logax) = x
– Thus, f(x) and g(x) are inverses of each other
• e.g. Let x = 2, f(x) = log2x and g(x) = 2x. Then
(f ◦ g)(2) = log2(22) = log24 = 2 = x AND
(g ◦ f)(2) = 2(log22) = 21 = 2 = x
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Basic Logarithmic Properties
Basic Logarithmic Properties
• logaa = 1 because a1 = a
– The logarithm base a of the same value is 1
• loga1 = 0 because a0 = 1, a ≠ 0
• loga(ax) = x and a(logax) = x because the
logarithm and exponential functions with base
a are inverses
• If logax = logay, then x = y
– This is known as the one-to-one property for
logarithms and will be very important when we
solve logarithmic equations
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Basic Logarithmic Properties
(Example)
Ex 3: Use the properties of logarithms to
evaluate:
a) log3 34
b) log 
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Graphing Logarithmic Functions
Logarithmic Functions as a Reflection
of Exponential Functions
• Given y = ax, we can create a table of values:
– By switching the x and y coordinates
in the table, we
obtain points for
logax
– Notice the graphs
reflected about
line y = x
y=
are
the
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Graphing Logarithmic Functions
• When graphing y = logax, it is easier to view
the function in its exponential form x = ay
– Pick values of y and calculate x
– Remember that the logarithmic and exponential
forms of a function are equivalent
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Properties of Graphs of Logarithmic
Functions
• Given the logarithmic function f(x) = logax
where a > 0 and a ≠ 1, the following properties
hold:
f(x) Is one-to-one
– Domain & range of f(x)
• Domain: (0, +oo); Range: (-oo, +oo)
– Contains the point (1, 0) and passes through the
point (a, 1)
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Properties of Graphs of Logarithmic
Functions (Continued)
– If a > 1, we obtain
something like on
the right
tha
• As x  +oo, f(x)  +oo
• As x  0, f(x)  -oo
– If 0 < a < 1, we obtain
something on
the right
like tha
• As x  +oo, f(x)  -oo
• As x  0, f(x)  +oo
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Graphing Logarithmic Functions
(Example)
Ex 4: Sketch the graph of the logarithmic
function f x  log3 x and then use
translations to sketch g x   log3 x  1  2
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Domain of Logarithmic Functions
Domain of Logarithmic Functions
• Recall that the domain of f(x) = logax is x > 0
• Thus for any function g(x), the domain of
f(x) = loga[g(x)] must satisfy g(x) > 0
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Domain of Logarithmic Functions
(Example)
Ex 5: Find the domain of the logarithmic
function:
a) f x   log2 2 x  5
b) g x  logx 2  4x
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Common & Natural Logarithms
Common Logarithm
• Common logarithm: written f(x) = log(x)
(without a base)  f(x) = log10x
– A logarithm without an explicit base has an
implied base of 10
– The log button on your calculator operates in only
base 10
– Can “trick” the calculator to evaluate in other
bases
• We will discuss this concept in a few lessons
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Natural Logarithm
• We discussed the special exponential function
f(x) = ex
• The inverse of f(x) = ex is x = ey  f(x) = logex
• Natural logarithm: written f(x) = ln(x) 
f(x) = logex
• The same logarithm
properties hold for
common and natural
logarithms
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ex and ln x as Inverses
• Recall that f(x) = logax and g(x) = ax where
a > 0, a ≠ 1, x > 0 are inverses
• So are their counterparts f(x) = ln x and
g(x) = ex:
(f ◦ g)(x) = ln (ex) = x
(g ◦ f)(x) = e(ln x) = x
• This fact will become very important to us
when we discuss solving exponential and
logarithmic equations
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Evaluating Common & Natural
Logarithms
Ex 6: Use a calculator to approximate:
a) 2 ln(0.75)
b) log(12.5)
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Summary
• After studying these slides, you should be able to:
– Understand the relationship between logarithmic and
exponential functions
– Be able to convert between logarithmic and exponential
functions
– Graph a logarithmic function
– State the domain of a logarithmic function
– Understand the concept of the common and natural
logarithms
• Additional Practice
– See the list of suggested problems for 3.2
• Next lesson
– Properties of Logarithms (Section 3.3)
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