Transcript 9-6 Multiplying a Polynomial by a Monomial

Logarithmic Functions &
Graphs, Lesson 3.2, page 388
Objective: To graph logarithmic
functions, to convert between
exponential and logarithmic equations,
and find common and natural
logarithms using a calculator.
DEFINITION

Logarithmic function – inverse of exponential
function
x
y
 If y = b , then the inverse is x = b
So y is the power which we raise b to in
order to get x.

Since we can’t solve this for y, we change it
to logarithmic form which is

y = logbx
Think of logs like this…
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 undefined.
Changing Exponential  Log


Log form => logb N = P
 Ex) log28 = 3
Think: A logarithm equals an
exponent!
Exponential form => bP = N
 Ex) 23 = 8
Examples of Conversion
Log Form: logbN = P
Exponential Form: bP = N
Log264 = 6
Log101000 = 3
Log416 = 2
25 = 32
104 = 10000
44 = 256
Rewrite the following exponential
expression as a logarithmic one.
( x  2)
3
7
a ) log 7 ( x  2)  3
b) log 3 ( x  2)  7
c) log 3 (7)  x  2
d ) log 3 ( x  2)  7
See Example 1, page 389.
Check Point 1.

Write each equation in its equivalent
exponential form:
A) 3 = log7x
B) 2 = logb25

C) log426 = y

See Example 2, page 389.
Check Point 2.

Write each equation in its equivalent
logarithmic form:
A) 25 = x
B) b3 = 27

C) e y = 33

See Example 3, page 389.
Check Point 3.

Evaluate:
A) log10 100

C) log36 6

B) log3 3
See page 390.
BASIC LOG PROPERTIES
 logb b = 1
 logb 1 = 0
INVERSE PROPERTIES OF LOGS
 logb bx = x
 blogbx = x
Examples
Check Point 4.
 A) log99
b) log8 1
Check Point 5:
 A) log7 78
b) 3log317
Graphs

Since exponential and logarithmic
functions are inverses of each other,
their graphs are also inverses.
logb b  x
x
b



logb x
x
Logarithmic function and
exponential function are
inverses of each other.
The domain of the exponential
function is all reals, so that’s the
domain of the logarithmic
function.
The range of the exponential
function is x>0, so the range of
the logarithmic function is y>0.
See Example 6, page 391.
Check Point 6:
x
 Graph f(x) = 3 and g(x) = log3 x in the
same rectangular coordinate system.
Graph f(x) = 3x.
x
y = f(x) = 3x
(x, y)
0
1
(0, 1)
1
3
(1, 3)
2
9
(2, 9)
3
27
(3, 27)
1
1/3 (1, 1/3)
2
1/9 (2, 1/9)
3
1/27 (3,1/27)
Now let’s add f(x) = log3x.
(Simply find the inverse of each point from
f(x)= 3x.)
f(x)= 3x
(0, 1)
(1, 3)
(2, 9)
(3, 27)
(1, 1/3)
(2, 1/9)
(3,1/27)
f ( x)  log3 x
See Characteristics of Graphs
of Logs on page 392.

See Table 3.4 on Transformations.
Graphing Summary

Logarithmic functions are inverses of
exponential functions. Easier if rewrite as
an exponential before graphing.
1. Choose values for 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.
Comparing Exponential and Logarithmic
Functions
Domain Restrictions for
Logarithmic Functions



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.
Remember, the question is: What POWER can I
raise the base to, to get this value?
DOMAIN RESTRICTION:
y  logb x, x  0
See Example 7, page 393.

Check Point 7: Find the domain of
f(x)=log4 (x-5).
Common Logarithms -- Intro



If no value is stated for the base, it is
assumed to be base 10.
log(1000) means, “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.
COMMON LOGARITHMS
A common logarithm is a log that uses
10 as its base.
 Log10 y is written simply as log y.
 Examples of common logs are
Log 100, log 50, log 26.2, log (1/4)
 Log button on your calculator is the
common log *

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 (4)
Find each of the following common
logarithms on a calculator.
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)
Natural Logarithms -- Intro




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  
 x
you get closer and closer to e.
logarithmic functions that involve base e are found
throughout nature
Calculators have a button “ln” which represents
the natural log.
log x  ln(x)
e
Natural Logarithms



Logarithms, 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) ln 0.0000245
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)