4.4 evaluate logarithms & graph logarithmic functions p. 251 what is a logarithm? how do you...
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4.4 Evaluate Logarithms & Graph Logarithmic Functions
p. 251What 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 by = 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 Exp. form
•log216 = 4
•log1010 = 1
•log31 = 0
•log10 .1 = -1
•log2 6 ≈ 2.585
•24 = 16•101 = 10•30 = 1•10-1 = .1•22.585 = 6
Evaluate
•log381 =
•Log5125 =
•Log4256 =
•Log2(1/32) =
•3x = 81•5x = 125•4x = 256•2x = (1/32)
4
34
-5
Evaluating logarithms now you try some!
•Log 4 16 = •Log 5 1 =•Log 4 2 =•Log 3 (-1) =• (Think of the graph of y=3x)
20
½ (because 41/2 = 2) undefined
You should learn the following general forms!!!
•Log b 1 = 0 because b0 = 1
•Log b b = 1 because b1 = b
•Log b bx = x because bx = bx
Natural logarithms
•log e x = ln x
•ln means log base e
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
b. ln 0.3
Check
8
.3
0.903089987
–1.203972804
100.903 8
0.3e –1.204
TornadoesThe wind speed s (in miles per hour) near the center of a tornado can be modeled by:
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.
93 log d + 65s =
Solution
= 93 log 220 + 65Write function.
93(2.342) + 65
= 282.806
Substitute 220 for d.
Use a calculator.
Simplify.
The wind speed near the tornado’s center was about 283 miles per hour.
ANSWER
93 log d + 65s =
• 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 AssignmentPage 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 b x is the inverse of
•f(x) = bx
•f(g(x)) = x and g(f(x)) = x•Exponential and log functions
are inverses and “undo” each other.
•So: g(f(x)) = logbbx = x• f(g(x)) = blog
bx = x
•10log2 = •Log39x =•10logx =•Log5125x =
2Log3(32)x =Log332x=2x
x3x
Use Inverse Properties
Simplify the expression.
a. 10log4 b. 5
log 25x
SOLUTION
Express 25 as a power with base 5.
a. 10log4 = 4
b. 5
log 25x = (52 ) x5
log
=5
log 52x
2x=
Power of a power property
blog xb = x
blog bx = x
Find Inverse PropertiesFind the inverse of the function.
SOLUTION
b.
a. y = 6 x b. y = ln (x + 3)
a.
6log
From the definition of logarithm, the inverse ofy = 6x is y= x.
Write original function.y = ln (x + 3)Switch x and y.x = ln (y + 3)Write in exponential form.Solve for y.
=ex (y + 3)=ex – 3 y
ANSWER The inverse of y = ln (x + 3) is y = ex – 3.
Use Inverse PropertiesSimplify the expression.
SOLUTION
10. 8 8log x
8 8log x = x
blog bb = x
11. 7
log 7–3x
SOLUTION
7log 7–3x = –3x
alog ax = x
Exponent form
Log form
Log form
Exponent form
Use Inverse PropertiesFind the inverse of14. y = 4x
SOLUTION
From the definition of logarithm, the inverse of
4log y = x.is
y = ln (x – 5).Find the inverse of15.
y = ln (x – 5)
SOLUTION
Write original function.Switch x and y.x = ln (y – 5)Write in exponential form.Solve for y.
=ex (y – 5)=ex + 5 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 y=3x • 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) Switch the x &
y
•ex = y + 1 Write in exp form
•ex – 1 = y 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 functionGraph the function.
SOLUTION
a. y = 3log x
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