3.2 logarithmic functions 2015 digital lesson. 3.1 warm-up mr. smith deposited $6,500 in an account...
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3.2Logarithmic Functions
2015
Digital Lesson
3.1 Warm-upMr. Smith deposited $6,500 in an account that pays
the account pays 4.5% interest, compounded monthly.
How much will he have in the account after 6 years?
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HWQ 9/8Mr. Jones received a bonus equivalent to
15% of his yearly salary and has decided to deposit
it in a savings account in which interest is compounded
continuously. His salary is $68,500 per year and
the account pays 5.5% interest.
How much interest will his deposit earn after 5 years?
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What is a log?
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2log 8 because
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For x 0 and 0 a 1, y = loga x if and only if x = a y.
The function given by f (x) = loga x is called the
logarithmic function with base a.
Every logarithmic equation has an equivalent exponential form: y = loga x is equivalent to x = a y
A logarithmic function is the inverse function of an exponential function.
Exponential function: y = ax
Logarithmic function: y = logax is equivalent to x = ay
A logarithm is an exponent!
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y = log2( )21 = 2
y
2
1
Examples: Write the equivalent exponential equation and solve for y.
1 = 5 yy = log51
16 = 4y y = log416
16 = 2yy = log216
SolutionEquivalent Exponential
Equation
Logarithmic Equation
16 = 24 y = 4
2
1= 2-1 y = –1
16 = 42 y = 2
1 = 50 y = 0
Converting between Forms
Log Form Exponential Form
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log10 –4 LOG –4 ENTER ERROR
no power of 10 gives a negative number
The base 10 logarithm function f (x) = log10 x is called the
common logarithm function.
The LOG key on a calculator is used to obtain common logarithms.
Examples: Calculate the values using a calculator.
log10 100
log10 5
Function Value Keystrokes Display
LOG 100 ENTER 2
LOG 5 ENTER 0.69897005
2log10( ) – 0.3979400LOG ( 2 5 ) ENTER
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Examples: Solve for x: log6 6 = x log6 6 = 1 property 2 x = 1
Simplify: log3 35
log3 35 = 5 property 3
Simplify: 7log7
9
7log7
9 = 9 property 3
Properties of Logarithms 1. loga 1 = 0 since a0 = 1. 2. loga a = 1 since a1 = a.
4. If loga x = loga y, then x = y. one-to-one property 3. loga a
x = x and alogax = x inverse property
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x
y
Graph f (x) = log2 x
Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x.
83
42
21
10
–1
–2
2xx
4
1
2
1
y = log2 x
y = xy = 2x
(1, 0)
x-intercept
horizontal asymptote y = 0
vertical asymptote x = 0
38
24
12
01
–2
log2 xx1
41
2–1
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Example: Graph the common logarithm function f(x) = log10 x.
by calculator
1
10
10.6020.3010–1–2f(x) = log10 x
10421x 1
100
y
x5
–5
f(x) = log10 x
x = 0 vertical
asymptote
(0, 1) x-intercept
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The graphs of logarithmic functions are similar for different values of a. f(x) = loga x (a 1)
3. x-intercept (1, 0)
5. increasing6. continuous7. one-to-one 8. reflection of y = a
x in y = x
1. domain ),0( 2. range ),(
4. vertical asymptote )(0 a s 0 xfxx
Graph of f (x) = loga x (a 1)
x
y y = x
y = log2 x
y = a x
domain
range
y-axisvertical
asymptote
x-intercept(1, 0)
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The function defined by f(x) = loge x = ln x
is called the natural logarithm function.
Use a calculator to evaluate: ln 3, ln –2, ln 100
ln 3ln –2ln 100
Function Value Keystrokes Display
LN 3 ENTER 1.0986122ERRORLN –2 ENTER
LN 100 ENTER 4.6051701
y = ln x
(x 0, e 2.718281)
y
x5
–5
y = ln x is equivalent to e y = x
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Properties of Natural Logarithms 1. ln 1 = 0 since e0 = 1. 2. ln e = 1 since e1 = e. 3. ln ex = x and eln x = x inverse property 4. If ln x = ln y, then x = y. one-to-one property
Examples: Simplify each expression.
2
1ln
e 2ln 2 e inverse property
2 0l ne 2 0 inverse property
eln3 3)1(3 property 2
00 1ln property 1
Finding the Domains of Logarithmic Functions
Find the Domain of each Function:
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) ln 2a f x x ) ln 2b f x x 2, , 2
y
x5
–5
y
x5
–5
Find the domain, vertical asymptotes, and x-intercept. Sketch a graph.
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6logf x xy
x5
–5
: 0,Domain
: 0VA x
int : 1,0x
Find the domain, vertical asymptotes, and x-intercept. Sketch a graph.
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10log 1 4g x x y
x5
–5
: 1,Domain
: 1VA x
4int : 1 10 ,0x
Homework
• Pg. 195 1-19 odd, 25-29 odd, 35-47 odd
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