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
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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
6
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
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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
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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)
x  log3 ( y  1)
3  y 1
x
3 1  y
x
1
Switch x and y
Re write as
exp onential
f ( x)  3  1
x
13
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
6
except everything
4
is rotated !
-10
2
-5
5
-2
-4
-6
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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
ln x  3
log  x  5 
log 3  x  2   1
log( x)
4
log (2)
ln  x  6 
log( x)  3
ln( x)  3
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!!!
x0
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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
y
 log( x  2) 
0
 log 3
Graph and label
asymptote : x  2
x  int : (3, 0)
D : (2, ) ; R(, )
Asymptote!!!
x2
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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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