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3-2 Logarithmic Functions Chapter 3 Exponential and Logarithmic Functions
3-2 Logarithmic FunctionsChapter 3 Exponential and Logarithmic Functions
Warm-up•Describe the transformation of f(x) that
results in the graph of g(x). Then sketch each graph.
1. 2.
3.
4
21)(;
21)(
xx
xgxf 52)(;2)( 2 xx xgxf
1)(;)( xx exgexf
Key Concept: Relating Logarithmic and Exponential Forms
•Exponential Form •Logarithmic Formyxb log xb y
base exponent base exponen
t
•When evaluating a logarithm, remember that
the
logarithm is the exponent.
•To evaluate a logarithm, change it to exponential form first. Then use what is known about exponents to simplify.
yxb log
Example 1: Evaluate each logarithma.
b.
c.
d.
16log2
1251log5
271log3
17log17
Key Concept: Basic Properties of Logarithms
01log b
01log b
1log bb
xb xb log
xb xb log
Example 2: Apply the Properties of LogarithmsEvaluate each expression:1.
2.
3.
4.
512log8
2.15log2222
81log9
1log33
1log33
Key Concept: Basic Properties of Common Logarithms•A logarithm with base 10 or log10 is called
a common logarithm, and it is often written without the base.
•The common logarithm function y = log x is the inverse of the exponential function y = 10x.
•The properties of common logarithms also
hold true for common logarithms.
Key Concept: Basic Properties of Common Logarithms
01log
110log
xx 10log
xx log10
Example 3: Evaluate each expression1. log 10,000
2. 10log 12
3. log 14 (use a calculator)
4. log (-11)
Key Concept: Basic Properties of Natural Logarithms
•A logarithm with the base e or loge is called a natural logarithm and is denoted ln.
•The natural logarithm function y = ln x is the inverse of the exponential function y = ex.
•The properties for logarithms also hold true for natural logarithms.
Key Concept: Basic Properties of Natural Logarithms
•ln 1 = 0
•ln e = 1
•ln ex = x
•e ln x = x
Example 4: Evaluate each expression1. ln e 4.6
2. ln (-1.2)
3. e ln 4
4. ln 7
Graphs of Logarithmic Functions?
Real World Example: Earthquakes•Richter Scale
• BTaR
log
Real World Example: Earthquakes•Richter Scale
• BTaR
logB
TaR
log
amplitude
Real World Example: Earthquakes•Richter Scale
• BTaR
logB
TaR
log
amplitude
Period of the seismic wave in seconds
Real World Example: Earthquakes•Richter Scale
• BTaR
logB
TaR
log
amplitude
Period of the seismic wave in seconds
A factor that accounts for
the weakening of seismic
waves
Real World Example: Earthquakes•Richter Scale
•
1. Find the intensity of an earthquake with an amplitude of 250 microns, a period of 2.1 seconds, and B = 5.4.
BTaR
logB
TaR
log
amplitude
Period of the seismic wave in seconds
A factor that accounts for
the weakening of seismic
waves
Real World Example: Earthquakes•Richter Scale
•
2. Earthquakes with an intensity of 6.1 or greater can cause considerable damage. Determine the amplitude of an earthquake whose intensity is 6.1 with a period of 3.5 seconds and B = 3.7.
BTaR
logB
TaR
log
amplitude
Period of the seismic wave in seconds
A factor that accounts for
the weakening of seismic
waves
Assignment: p. 178-9•1 – 23 odds, 27
Be sure to show the
set-up used to
calculate this one.
