Download algebra ii - MooreMath23

ALGEBRA II
LESSON 10-2
Pg. 531-536
LOGARITHMS &
LOGARITHM FUNCTIONS
Introduction to Logarithms: Who can tell the class what a decibel is?
It is a measure of sound, but is 40 dB half as much as 80 dB? NO
What about earthquakes? How are they measured for intensity? Richter Scale
Is an earthquake of registering 6 on the scale only twice as much as one of 3?
What about tornados? Is an F2 half as intense as an F4?
Today’s Key Concepts: We should know what an exponential function is…so now a logarithmic function is
the inverse of the exponential function. If we raise 102 by one exponential power what is the end result?
Here are the powers of 10 and their logarithms:
Powers of 10 :
1/1000
1/100 1/10
1
10
100
1000
10,000
Logarithms:
-3
-2
0
1
2
3
4
-1
then the system of common logarithms -- base 10 -- is implied.
The system of common logarithms has 10 as its base when the base is not indicated:
log 100 = 2
102 = 100
Logarithms replace a geometric series with an arithmetic series.
_______________________________________________________________________
Logarithms are the "opposite" of exponentials, just as subtraction is the opposite of addition
and division is the opposite of multiplication. Logs "undo" exponentials. Technically speaking, logs
are the inverses of exponentials. In practical terms, I have found it useful to think of logs in terms of
The Relationship:
—The Relationship—
..............is equivalent
x
y=b
to...............
logb(y) = x
(means the exact same
thing as)
On the left-hand side above is the exponential statement "y = bx". On the right-hand side above,
"logb(y) = x" is the equivalent logarithmic statement, which is pronounced "log-base-b of y equals x";
"b" is called "the base of the logarithm", just as b is the base in the exponential expression "bx".
Note that the base in both the exponential equation and the log equation (above) is "b",
but that the x and y switch sides when you switch between the two equations.
Here’s the way to easily transition from one to the other:
bx = y becomes log b y = x
*** Think of this: x and y are immediately next to the equal sign, they switch places and b gets
rolled by the log and gets small
___________________________________________________________
Definition of Exponential Function (exists in only quadrants I and II)
Graph of Logarithmic Function
Since x > 0, the graph of the above function will be in quadrants I and IV.
Comments on Logarithmic Functions

The exponential equation
could be written in terms of a logarithmic equation as
.



The exponential equation
can be written as the logarithmic equation
.
Since logarithms are nothing more than exponents, you can use the rules of exponents with
logarithms.
Logarithmic functions are the inverse of exponential functions. For example if (4, 16) is a point on
the graph of an exponential function, then (16, 4) would be the corresponding point on the graph
of the inverse logarithmic function.
Property 1: loga1 = 0 because a0 = 1.
Example 1: In the equation 140 = 1, the base is 14 and the exponent is 0. Remember that a
logarithm is an exponent, and the corresponding logarithmic equation is log14 1 = 0 where the 0 is
the exponent.
Example 2: In the equation (1/2)0 = 1, the base is ½ and the exponent is 0. Remember that
a logarithm is an exponent, and the corresponding logarithmic equation is log ½ 1 = 0
Example 3: Use the exponential equation x0 = 1 to write a logarithmic equation. The base
x is greater than 0 and the exponent is 0. The corresponding logarithmic equation is logx 1 = 0
.
Property 2: loga a = 1 because a1 = a.
Example 4: In the equation 31 = 3, the base is 3, the exponent is 1, and the answer is 3.
Remember that a logarithm is an exponent, and the corresponding log equation is log33 = 1.
Example 5: Use the exponential equation x1 = x to write a logarithmic equation. If the
base x is greater than 0, then logx x = 1.
Property 3: loga ax = x because ax = ax.
Example 6: Since you know that 34 = 34, you can write the logarithmic equation with base
3 as log3 34 = 4.
Example 7: Use the exponential equation 42 = 16 to write a logarithmic equation with
base 4. You can convert the exponential equation to the logarithmic equation log4 16 = 2. Since
the 16 can be written as 42 , the equation log4 16 = 2 can be written log4 42 = 2
.
RECAP LOGARITHMS:
Definition of Logarithm: logbx = n means bn = x.
Example 1. Write in exponential form: log232 = 5
Answer. 25 = 32
Example 2. Write in logarithmic form: 4−2 = 1
16
Answer. log4 1 = −2
16
The above rules are the same for all positive bases. The most common base is base 10.
Logarithms with a base 10 are called common logarithms. On your calculator, the base 10 logarithm
is noted by log.
There are an infinite number of bases and only a few buttons on your calculator. You can convert a
logarithm with a base that is not 10 but we will learn that in the next lesson.