4.4 Evaluate Logarithms &
Graph Logarithmic Functions
p. 251
What is a logarithm? How do you read it?
What relationship exists between logs
and exponents? What is the definition?
How do you rewrite log equations?
What are two special log values?
What is a common log? A natural log?
What logs can you evaluate using a
calculator?
Evaluating Log Expressions
• We know 22 = 4 and 23 = 8
• But for what value of y does 2y = 6?
• Because 22<6<23 you would expect
the answer to be between 2 & 3.
• To answer this question exactly,
mathematicians defined logarithms.
Definition of Logarithm to base b
• Let b & x be positive numbers & b ≠ 1.
• The logarithm of x with base a is
denoted by logbx and is defined:
• logbx = y if
y
b
=x
• This expression is read “log base b of x”
• The function f(x) = logbx is the
logarithmic function with base b.
• The definition tells you that the
equations logbx = y and by = x are
equivalent.
• Rewriting forms:
• To evaluate log3 9 = x ask yourself…
• “Self… 3 to what power is 9?”
• 32 = 9 so……
log39 = 2
Log form
• log216 = 4
• log1010 = 1
• log31 = 0
• log10 .1 = -1
• log2 6 ≈ 2.585
Exp. form
• 24 = 16
1
• 10 = 10
0
•3 = 1
• 10-1 = .1
• 22.585 = 6
Evaluate
x
•3
= 81
• log381 = 4
x
• 5 = 125
• Log5125 = 3
x = 256
•
4
• Log4256 = 4
x
• Log2(1/32) = -5 • 2 = (1/32)
Evaluating logarithms now you try
some!
• Log 4 16 = 2
• Log 5 1 = 0
• Log 4 2 = ½ (because 41/2 = 2)
• Log 3 (-1) = undefined
• (Think of the graph of y=3x)
You should learn the following
general forms!!!
0
b
• Log b 1 = 0 because
=1
1
• Log b b = 1 because b = b
• Log b bx = x because bx = bx
Natural logarithms
•log e x = lnex
• ln means log base e
Common logarithms
•log 10 x = log x
• Understood base 10 if
nothing is there.
Common logs and natural logs with
a calculator
log10 button
ln button
**Only common log and
natural log bases are on a
calculator.
Keystrokes
Expression Keystrokes
Display
a. log 8
0.903089987 100.903
b. ln 0.3
8
.3
Check
8
–1.203972804
e –1.204
0.3
Tornadoes
The wind speed s (in
miles per hour) near the
center of a tornado can be
modeled by:
s = 93 log d + 65
where d is the distance (in
miles) that the tornado
travels. In 1925, a tornado
traveled 220 miles through
three states. Estimate the
wind speed near the
tornado’s center.
Solution
s = 93 log d + 65
= 93 log 220 + 65
93(2.342) + 65
= 282.806
Write function.
Substitute 220 for d.
Use a calculator.
Simplify.
ANSWER
The wind speed near the tornado’s center was
about 283 miles per hour.
• What is a logarithm? How do you read it?
A logarithm is another way of expressing an
exponent. It is read log base b of y.
• What relationship exists between logs and
exponents? What is the definition?
• logax = y if ay = x
• How do you rewrite logs?
The base with the exponent on the other side of
the = .
• What are two special log values?
Logb1=0 and logbb=1
• What is a common log? A natural log?
Common log is base 10. Natural log is base e.
• What logs can you evaluate using a calculator?
Base 10
4.4 Assignment
Page 255, 3-6, 8-16,
20-26 even
4.4 Day 2
• How do you use inverse properties with
logarithms?
• How do you graph logs?
• g(x) = log
• f(x) = bx
b
x is the inverse of
• f(g(x)) = x and g(f(x)) = x
• Exponential and log functions
are inverses and “undo” each
other.
• So: g(f(x)) = logb
•
x
b
=x
log
x
f(g(x)) = b b = x
• 10log2 = 2
x
• Log39 = Log3(32)x =Log332x=2x
logx
• 10
= x
x
• Log5125 = 3x
Use Inverse Properties
Simplify the expression.
b. log 5 25x
a. 10log4
SOLUTION
a.
10log4
=4
log bx
b
=x
x
x
2
(
)
log
log
b.
Express 25 as a power with base 5.
5 5
5 25 =
Power of a power property
= log 5 52x
= 2x
logb bx = x
Find Inverse Properties
Find the inverse of the function.
a.
y=6
x
b. y = ln (x + 3)
SOLUTION
a. From
the definition of logarithm, the inverse of
x
y = 6 is y= log 6 x.
b. y = ln (x + 3)
x = ln (y + 3)
ex = (y + 3)
ex – 3 = y
Write original function.
Switch x and y.
Write in exponential form.
Solve for y.
ANSWER The inverse of y = ln (x + 3) is y = ex – 3.
Use Inverse Properties
Simplify the expression.
10. 8 log 8 x
SOLUTION
8
11.
log 8 x
b
log bb
=x
Exponent form
= x
log 7
Log form
7–3x
SOLUTION
log 7 7–3x = –3x
log a ax = x
Log form
Exponent form
Use Inverse Properties
x
14. Find the inverse of y = 4
SOLUTION
From the definition of logarithm, the inverse of
is y = log4 x.
15. Find the inverse of y = ln (x – 5).
SOLUTION
y = ln (x – 5)
Write original function.
x = ln (y – 5)
Switch x and y.
ex = (y – 5)
Write in exponential form.
ex + 5 = y
Solve for y.
ANSWER The inverse of y = ln (x – 5) is y = e x + 5.
Finding Inverses
• Find the inverse of:
• y = log3x
• By definition of logarithm, the inverse is
x
y=3
• OR write it in exponential form and
switch the x & y!
3y = x
3x = y
Finding Inverses cont.
• Find the inverse of :
• Y = ln (x +1)
• X = ln (y + 1)
• ex = y + 1
• ex – 1 = y
Switch the x & y
Write in exp form
solve for y
4.4 Graphing Logs
p. 254
Graphs of logs
• y = logb(x-h)+k
• Has vertical asymptote x=h
• The domain is x>h, the range is
all reals
• If b>1, the graph moves up to
the right
• If 0<b<1, the graph moves down
to the right
Graphing a log function
Graph the function.
a. y = log3 x
SOLUTION
Plot several convenient
points, such as (1, 0), (3, 1),
and (9, 2). The y-axis is a
vertical asymptote.
From left to right, draw a curve that starts
just to the right of the y-axis and moves up
through the plotted points, as shown below.
Graph y = log1/3x-1
• Plot (1/3,0) & (3,-2)
• Vert line x=0 is asy.
• Connect the dots
X=0
Graph y =log5(x+2)
• Plot easy points
(-1,0) & (3,1)
• Label the
asymptote x=−2
• Connect the
dots using the
asymptote.
X=-2
• How do you use inverse properties with
logarithms?
Exponential and log functions are inverses
and “undo” each other.
• How do you graph logs?
Pick 1, the base number, and a power of
the base for x.
4.4 Assignment Day 2
• Page 256, 28-44 even,
45-51 odd