3.3 logarithmic functions and their graphs we learned that, if a function passes the horizontal line...
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3.3 Logarithmic Functions and Their Graphs• We learned that, if a function passes the horizontal
line test, then the inverse of the function is also a function.
• So, an exponential function f(x) = bx, has an inverse that is a function.– This inverse is the logarithmic function with base b,
denoted or .log ( )b x logb x
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Changing Between Logarithmic and Exponential Form
• If x > 0 and 0 < b ≠ 1, then
if and only if .
• This statement says that a logarithm is an exponent. Because logarithms are exponents, we can evaluate simple logarithmic expressions using our understanding of exponents.
log ( )by x yb x
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Evaluating Logarithmsa.) because .
b.) because .
c.) because
d.) because
e.) because
2log 8 3 32 8
3log 3 1 2 1 23 3
5
1log 2
25 2
2
1 15 .
5 25
4log 1 0 04 1.
7log 7 1 17 7.
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Basic Properties of Logarithms• For 0 < b ≠ 1 , x > 0, and any real number y,
logb1 = 0 because b0 = 1.
logbb = 1 because b1 = b.
logbby = y because by = by.
becauselogb xb x log log .b bx x
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Evaluating Logarithmic and Exponential Expressions
(a)
(b)
(c)
2log 8 32log 2 3.
3log 3 1 23log 3 1 2.
6log 116 11.
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Common Logarithms – Base 10• Logarithms with base 10 are called common
logarithms.• Often drop the subscript of 10 for the base when
using common logarithms.• The common logarithmic function:
y = log x if and only if 10y = x.
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Basic Properties of Common Logarithms• Let x and y be real numbers with x > 0.
– log 1 = 0 because 100 = 1.
– log 10 = 1 because 101 = 10.
– log 10y = y because 10y = 10y .
– because log x = log x.
• Using the definition of common logarithm or these basic properties, we can evaluate expressions involving a base of 10.
log10 x x
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Evaluating Logarithmic and Exponential Expressions – Base 10
(a)
(b)
(c)
(d)
log100 10log 100 2
5log 10 1 5log10 1
5
1log1000 3
1log10
3log10 3log610 6
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Evaluating Common Logarithms with a Calculator
• See example 4 p. 303
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Solving Simple Logarithmic Equations• Solve each equation by changing it to exponential
form:a.) log x = 3 b.)
a.) Changing to exponential form, x = 10³ = 1000.
b.) Changing to exponential form, x = 25 = 32.
2log 5x
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Natural Logarithms – Base e
• Because of their special calculus properties, logarithms with the natural base e are used in many situations.
• Logarithms with base e are natural logarithms.– We use the abbreviation “ln” (without a subscript)
to denote a natural logarithm.
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Basic Properties of Natural Logarithms• Let x and y be real numbers with x > 0.– ln 1 = 0 because e0 = 1.
– ln e = 1 because e1 = e.
– ln ey = y because ey = ey.
– eln x = x because ln x = ln x.
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Evaluating Logarithmic and Exponential Expressions – Base e
(a)
(b)
(c)
ln e
5ln e
ln 4e
loge e 1/ 2
5loge e 5
4
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Transforming Logarithmic Graphs• Describe how to transform the graph of y = ln x or y
= log x into the graph of the given function.a.) g(x) = ln (x + 2)
The graph is obtained by translating the graph of y = ln (x) two units to the LEFT.
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Transforming Logarithmic Graphs• Describe how to transform the graph of y = ln x or y
= log x into the graph of the given function.b.) h(x) = ln (3 - x)
The graph is obtained by applying, in order, a reflection across the y-axis followed by a transformation three units to the RIGHT.
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Transforming Logarithmic Graphs• Describe how to transform the graph of y = ln x or y
= log x into the graph of the given function.c.) g(x) = 3 log x
The graph is obtained by vertically stretching the graph of f(x) = log x by a factor of 3.
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Transforming Logarithmic Graphs• Describe how to transform the graph of y = ln x or y
= log x into the graph of the given function.d.) h(x) = 1+ log x
The graph is obtained by a translation 1 unit up.
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Measuring Sound Using Decibels• The level of sound intensity in decibels (dB) is
where (beta) is the number of decibels, I is the sound intensity in W/m², and I0 = 10 – 12 W/m² is the threshold of human hearing (the quietest audible sound intensity).
010log( / ),I I
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More Practice!!!!!
• Homework – Textbook p. 308 #2 – 52 even.