Download y log x, x 0 - Shelton State

Objectives
– Change from logarithmic to exponential form.
– Change from exponential to logarithmic form.
– Evaluate logarithms.
– Use basic logarithmic properties.
– Graph
G h logarithmic
l
ith i ffunctions.
ti
– Find the domain of a logarithmic function.
– Use common logarithms.
– Use natural logarithms.
Logarithmic Functions
Section 3.2
Inverse of y = 3x
Flashback
Consider the graph of the exponential function
y = f(x) = 3x.
Begin with
Interchange variables
• Is f(x) one-to-one?
• Does f(x) have an
i
inverse
th
thatt is
i a
function?
• Find the inverse.
y = 3x
x = 3y
Now, solve for y.
y= the
th power to
t which
hi h 3 mustt be
b raised
i d in
i order
d
to obtain x.
Symbolically, y = log 3 x
“The logarithm, base 3, of x.”
Domain Restrictions for Logarithmic Functions
Logarithm
For all positive numbers b, where b ≠ 1,
b y =x
is equivalent to
≠
y=logb x
Logbx is an exponent to which the base b
must be raised to obtain x.
• Since a positive number raised to an exponent (positive or
negative) always results in a positive value, you can ONLY
take the logarithm of a POSITIVE NUMBER.
• Remember, the question is: What POWER can I raise the
base to, to get this value?
• DOMAIN RESTRICTION:
y = logb x, x > 0
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Write on top of your test!!
log a x = y ↔ x = a y
A logarithm is an exponent!
Common Logarithms
Logarithms, base 10, are called common
logarithms.
For all p
positive numbers x,
lo g x
=
lo g 1 0 x
Log button on your calculator
is the common log.
Example
• Find each of the following common logarithms on a
calculator.
Round to four decimal places.
a) log 723,456
b) log 0.0000245
c)) log
l ((−4)
4)
Natural logarithms
• Logarithms, base e, are called natural logarithms.
• ln(x) represents the natural log of x, which has a
base=e
x
⎛ 1⎞
• What is e? If you plug large values into ⎜1 + ⎟
yyou g
get closer and closer to e.
⎝ x⎠
• logarithmic functions that involve base e are found
throughout nature
• Calculators have a button “ln” which represents the
natural log.
log e x = ln( x)
Example
• Find each of the following natural logarithms on a
calculator.
Round to four decimal places.
a) ln 723,456
b) ln 0.0000245
c) ln (−4)
Logarithmic Conversions
• Convert each of the following to a
logarithmic equation.
a) 25 = 5x
b) ew = 30
2
Example
• Convert each of the following to an
exponential equation.
Rewrite the following exponential
expression as a logarithmic one.
3( x + 2 ) = 7
1) log 7 ( x + 2) = 3
a) log7 343 = 3
2) log 3 ( x + 2) = 7
b) logb R = 12
3) log 3 (7) = x + 2
4) log 3 ( x − 2) = 7
Finding Logarithms
• Find each of the following logarithms
without using a calculator.
a) log2 16
b) log10 1000
c) log16 4
d) log10 0.001
Summary of Properties of Logarithms
F o r a > 0 , a ≠ 1 ,a n d a n y re a l n u m b e r k ,
1 . lo g a a = 1 , ln e = 1
2 . lo g a 1 = 0 , ln 1 = 0
A d d itio n a l L o g a rith m ic P ro p e rtie s
3 . lo g a a k = k
1
1
=
= 10−3.
1000 103
Example
Evaluate each expression without using a calculator.
a.) log6 6
b.)log11 11
c.)ln1
c
)ln1
d.)log15 1
e.)log4 46
4 . a lo g a k = k , k > 0
Logarithmic Functions
• Logarithmic functions are inverses of exponential
functions.
Graph y = log 3 x by converting to its equivalent
exponential form: x = 3y
1. Choose values for y.
y
2. Compute values for x.
3. Plot the points and connect them with a
smooth curve.
* Note that the curve does not touch or cross
the y-axis.
3
Logarithmic Functions continued
Graph: x = 3y
y = log 3 x
x = 3y
y
(x, y)
1
0
(1, 0)
3
1
(3 1)
(3,
9
2
(9, 2)
1/3
−1
(1/3, −1)
1/9
−2
(1/9, −2)
1/27
−3
(1/27, −3)
Comparing Exponential and Logarithmic Functions
Logarithmic Functions
Remember: Logarithmic functions are inverses
of exponential functions.
Th iinverse off f(x)
The
f( ) = a x
is given by f -1(x)=loga x
Asymptotes
• Recall that the horizontal asymptote of
the exponential function y = ax is the
x-axis.
• The horizontal axis of a logarithmic
function y=loga x is the y-axis.
4
Graphs of Logarithmic Functions
• Graph: y = f(x) = log6 x.
– Select y.
– Compute x.
x, or 6 y
1
6
36
216
1/6
1/36
Transformations of logarithmic functions
are treated as other transformations.
• Follow order of operations.
• Note: When graphing a logarithmic
function,, the graph
g p only
y exists for x>0.
• WHY?
y
0
1
2
3
−1
−2
– If a positive number is raised to an exponent,
no matter how large or small, the result will
always be POSITIVE!
Example
• Graph each of the following.
• Describe how each graph can be obtained from the
graph of y = ln x.
• Give the domain and the vertical asymptote of
each function.
• a) f(x) = ln (x − 2)
• b) f(x) = 2 − ¼ ln x
• c) f(x) = |ln (x + 1)|
Graph f(x) = 2 − ¼ ln x
• The graph is a vertical shrinking
by a factor of 14 , followed by a
reflection across the x-axis, and
then a translation up 2 units.
• The domain is the set of all
positive real numbers.
• The y-axis is the vertical
asymptote
asymptote.
x
0.1
0.5
1
3
5
f(x)
2.576
2.173
2
1.725
1.598
Graph f(x) = ln (x − 2)
• The graph is a shift 2 units
right.
• The domain is the set of all
real numbers greater than 2.
• The line x = 2 is the vertical
asymptote.
x
f( )
f(x)
2.25
−1.386
2.5
−0.693
3
0
4
0.693
5
1.099
Graph f(x) = |ln (x + 1)|
• The graph is a translation 1 unit to
the left.
Then the absolute value has the
effect of reflecting negative outputs
across the x-axis.
• The domain is the set of all real
numbers greater than −1.
• The line x = −1 is the vertical
asymptote.
x
−0.5
0
1
3
6
f(x)
0.693
0
0.693
1.386
1.946
5
Application: Walking Speed
• In a study by psychologists Bornstein and Bornstein,
it was found that the average walking speed w, in
feet per second, of a person living in a city of
population P, in thousands, is given by the function
w(P) = 0.37 ln P + 0.05.
The population of Salem
Salem, Oregon is about 137
137, 000
000.
Find the average walking speed of people living in
Salem.
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