Review
Logarithmic Functions &
Graphs
„
Find the inverse of y = 5x.
„
Graph the function and its inverse
inverse.
y
10
Objective: To graph logarithmic
functions, to convert between
exponential and logarithmic equations,
and find common and natural
logarithms using a calculator.
x
-10
-10
DEFINITION
„
„
„
Think of logs like this…
Logarithmic function – __________ of
exponential function
bx,
If y =
then the inverse is x =
Since we can’t solve this for y, we
change it to logarithmic form which is
„
b y.
__________
Changing Exponential ↔ Log
„
„
10
Log form => logb N = P
„ Ex) log28 = 3
Think: A logarithm __________ an
exponent!
Exponential form => bP = N
„ Ex) 23 = 8
logbN = P
„
„
and
bp = N
Key: b = base, N = number, P = power
Restrictions:
b > 0 and b cannot equal 1
*N > 0 because the log of zero or a
negative number is __________.
Examples of Conversion
Log Form: logbN = P
Exponential Form: bP = N
Log264 = 6
Log101000 = 3
Log416 = 2
25 = 32
104 = 10000
44 = 256
1
Rewrite the following exponential
expression as a logarithmic one.
3( x + 2 ) = 7
a ) log 7 ( x + 2) = 3
b) log 3 ( x + 2) = 7
Example – Log to Exponential
„
Write each equation in its equivalent
exponential form:
A) 3 = log7x
B) 2 = logb25
„
C) log426 = y
„
c) log 3 (7) = x + 2
d ) log 3 ( x − 2) = 7
Example – Exponential to Log
„
Write each equation in its equivalent
logarithmic form:
A) 25 = x
B) b3 = 27
„
C) e y = 33
„
Properties
BASIC LOG PROPERTIES
„ logb b = ____
„ logb 1 = ____
INVERSE PROPERTIES OF LOGS
„ logb bx = ____
„ blogbx = ____
Example
„
Evaluate:
A) log10 100
„
C) log36 6
„
B) log3 3
Examples
„
a) log99
b) log8 1
„
c) log7 78
d) 3log317
2
Graphs
„
Example
Since exponential and logarithmic
functions are inverses of each other,
their graphs are also inverses.
„
Graph f(x) = 3x and g(x) = log3 x in the
same rectangular coordinate system.
Now let’s add f(x) = log3x.
Graph f(x) = 3x.
x
y = f(x) = 3x
(Simply find the inverse of each point from
f(x)= 3x.)
(x, y)
f(x)= 3x
0
1
(0, 1)
(0, 1)
1
3
(1 3)
(1,
(1 3)
(1,
2
9
(2, 9)
(2, 9)
3
27
(3, 27)
(3, 27)
−1
1/3 (−1, 1/3)
−2
1/9 (−2, 1/9)
(−2, 1/9)
1/27 (−3,1/27)
(−3,1/27)
−3
f(x) = log3x
(−1, 1/3)
Comparing Logarithmic and Exponential Functions
Graphing Summary
„
Logarithmic functions are inverses of
__________ functions.
1 Choose values for y.
1.
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
Transformation of logarithmic
functions is treated as other
transformations.
„
„
Domain Restrictions for
Logarithmic Functions
„
Follow __________ of operations.
Note: When graphing a logarithmic
function, the graph only exists for
__________, WHY? If a positive
number is raised to an exponent, no
matter how large or small, the result
will always be POSITIVE!
„
„
y = log b x, x > 0
Example
„
Common Logarithms -- Intro
Find the domain of f(x)=log4 (x-5).
„
„
„
A common logarithm is a log that uses
10 as its base.
„ Log10 y is written simply as
__________.
„ Examples of common logs are
Log 100, log 50, log 26.2, log (1/4)
„ Log button on your calculator is the
common log *
If no value is stated for the base, it is
assumed to be base _____.
log(1000) means
means, “What
What power do I raise
10 to, to get 1000?” The answer is 3.
log(1/10) means, “What power do I raise 10
to, to get 1/10?” The answer is -1.
Find each of the following common
logarithms on a calculator.
COMMON LOGARITHMS
„
Since a positive number raised to an exponent
(pos. or neg.) always results in a positive value,
you can ONLY take the logarithm of a POSITIVE
NUMBER.
NUMBER
Remember, the question is: What POWER can I
raise the base to, to get this value?
DOMAIN RESTRICTION:
„
Round to four decimal places.
a) log 723,456
b) llog 0
0.0000245
0000245
c) log (−4)
Function Value
Readout
Rounded
log 723,456
5.859412123
5.8594
log 0.0000245
−4.610833916
−4.6108
ERR: non real ans
Does not exist
log (−4)
4
Natural Logarithms
Natural Logarithms -- Intro
„
„
ln(x) represents the __________ log of x, which
has a base of e
Logarithmic functions that involve base e are
found throughout nature.
log e x = ln( x )
„
Logarithms with base e are called
natural logarithms.
„
The abbreviation “ln” is generally used for
natural logarithms.
Thus,
ln x means loge x.
„
* ln button on your calculator
is the natural log *
Find each of the following natural
logarithms on a calculator.
„
Round to four decimal places.
a) ln 723,456
b) lln 0
0.0000245
0000245
c) ln (−4)
Function Value
Readout
Rounded
ln 723,456
13.49179501
13.4918
ln 0.0000245
−10.61683744
−10.6168
ERR: nonreal
answer
Does not exist
ln (−4)
5