Graphs of Logarithmic Functions
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Transcript Graphs of Logarithmic Functions
Algebra 2: Section 8.4
Logarithmic Functions
(Day 1)
1
Solving for “x”
Addition
x–3=5
Subtraction
3+x=9
Multiplication
1/2x = 4
Division
5x = 25
Power
x3 = 27
3
Roots
x 4
If “x” is an exponent?
2
Definition of Logarithm
log b y x iff
b y
x
logby is read as “log base b of y”
3
Examples
Rewrite the equations in
exponential form.
Logarithmic Function
Exponential Function
1. log39 = 2
1. 32 = 9
2. log81 = 0
2. 80 = 1
3. log5 (1/25) = -2
3. 5-2 = 1/25
4
Examples
Evaluate the expressions.
Hint: For logby ask yourself what
power of b gives you y?
4. log464
What power of 4 gives you 64?
4x = 64
5. log20.125
Answer: 3
What power of 2 gives you 0.125?
2x = 0.125
Answer: -3
6. log1/4256
What power of ¼ gives you 256?
1/4x = 256
7. log322
Answer: -4
What power of 32 gives you 2?
32x = 2
5
Answer: 1/5
Common and Natural Logs
Common Logarithm
(the base of 10 is not written)
log10x = log x
Natural Logarithm
(remember “e” = natural base)
logex = ln x
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Examples
Evaluate:
(Round to 3 decimals)
8. log 7
= 0.845
9. ln 0.25
= -1.386
On TI-83:
LOG button is
base 10 and is to
the left of 7
LN button is
base e and is to
the left of 4
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Logarithm Inverse Properties
g ( x) log b x and f ( x) b
x
are inverses of each other!
This means that...
log b b x and b
x
logb x
x
8
Examples
log b b x
x
Simplify the expressions.
10. 20
log 20 x
11. log 4 4
12. 10
b
=x
2
log b x
x
2
5
log 5
x
3x
13. log 5 125 = log 5 5
= 3x
9
Finding Inverses of Logarithms
SAME Steps as Before!!!
First, switch the x’s and y’s.
Rewrite the logarithm equation as an
exponential equation.
Solve for y.
10
Examples
Find the inverse of the following
functions.
y = log8 x Switch x and y
x = log8y Re write as exp onential
8x = y
y = 8x
14.
11
Examples
y = ln (x – 10) Switch x and y
x = ln(y – 10)
Re write as
x
e = y – 10
exp onential
y = ex + 10
15.
12
16. f ( x) log3 ( x 1)
y log3 ( x 1)
Switch x and y
x log3 ( y 1)
Re write as
exp onential
3 y 1
x
3 1 y
x
1
f ( x) 3 1
x
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Homework
p.490
#16-64 evens
14
Algebra 2: Section 8.4
Logarithmic Functions
(Day 2)
(Graphing…yeah!)
15
Definition of Logarithm
(Reminder)
log b y x iff b y
SAME AS
x
x log b y iff b y
x
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Change of Base Formula
Used to evaluate logs that are bases other
than 10 or e.
log u
ln u
log c u
or
log c
ln c
Or to punch logs of base other than 10 or
e into the calculator (for graphing).
log x
log c x
log c
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Graphs of Logarithmic Functions
8
SAME AS " e " graphs
except everything
is rotated !
-10
6
4
2
-5
5
-2
-4
-6
10
Graphs of Logarithmic Functions
y = logb(x – h) + k
Asymptote: h (x = “h”)
Domain: ( h, )
Range: ( , )
If b>1, curve opens up
If 0<b<1, curve opens down
•
•
•
To graph:
Show the asymptote
Plot the x-intercept (calc or…..)
find by setting y = 0 (will have to do for SEVERAL!!!)
rewrite as an exponential equation
Solve for x
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How to write in Calculator
log 2 x 4
ln x 6
log x 3
log( x)
4
log (2)
log( x) 3
ln( x ) 3
ln x 3
log x 5
log 3 x 2 1
ln x 6
log x 5
log x 2
log(3)
1
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Examples
State the asymptote, the domain,
Does
therange
graph
to ?
and the
of disappear
each function.
1.
y = log1/2x + 4
Curve opens down
y log1/ 2 ( x 0) 4
asymptote : x 0
x int : (16, 0)
D : (0, ) ; R(, )
y
log x
1
log
2
4
Graph and label
Asymptote!!!
x0
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Does the graph
disappear at x 2?
Examples
2. y = log3(x – 2)
Curve opens up
y log3 ( x 2) 0
asymptote : x 2
x int : (3, 0)
D : (2, ) ; R(, )
y
log( x 2)
0
log 3
Graph and label
Asymptote!!!
x2
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Homework
p.491
#65-76 all
State asymptote, x-intercept, domain, range
Be sure asymptotes are graphed and labeled
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