Download Logarithmic Functions

Int. Alg. Notes
Section 9.3
Page 1 of 8
Section 9.3: Logarithmic Functions
Big Idea: The logarithmic function is the inverse of the exponential function.
Big Skill: You should be able to convert between exponential and logarithmic expressions, and graph
logarithmic functions.
The logarithmic function is the inverse of the exponential function. To understand this, let’s make an analogy
to power functions and their inverses (which are roots) :
Power Functions and Their Inverses
If x  16 (and x  0), then
2
x 2  16
x  16
x4
We know this is the correct answer because 42 = 16.
The square root found the base, that when squared, gives an answer of 16.
The square root is the inverse of the power function x2: If f  x   x2 , then f 1  x   x .
Notice that there is no index written for a square root; it is an agreed-upon notation that a square root “undoes”
a 2nd power.
If x 3  27 , then
3
x3  3 27
x  3 27
x3
We know this is the correct answer because 33 = 27.
The cube root found the base, that when cubed, gives an answer of 27.
The cube root is the inverse of the power function x3: If f  x   x3 , then f 1  x   3 x .
Notice that we had to write an index of 3 in the radical to indicate it “undoes” a 3rd power.
If x 4  625 (and x  0), then
4
x 4  4 625
x  4 625
x5
We know this is the correct answer because 54 = 625.
The fourth root found the base, that when raised to the fourth power, gives an answer of 625.
The fourth root is the inverse of the power function x4: If f  x   x4 , then f 1  x   4 x .
Notice that we had to write an index of 4 in the radical to indicate it “undoes” a 4th power.
If f  x   xn , then f 1  x   n x .
Notice: the notation for the inverse includes the power of the function.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 9.3
Page 2 of 8
Definition: Logarithmic Function
The logarithmic function to the base a, where a > 0 and a  1, is denoted by y  loga  x  (read as “y is the
logarithm to the base of a of x”, or as “y is the exponent on a that gives an answer of x”), and is defined as
x  ay .
y  loga  x  is equivalent to
Exponential Functions and Their Inverses
If 2  128 , then
log 2  2 x   log 2 128 
x
x  log 2 128 
x7
We know this is the correct answer because 27 = 128
The logarithm to the base of 2 found the exponent, that when placed on a base of 2, gives an answer of 128.
The logarithm to the base of 2 is the inverse of the exponential function 2x: If f  x   2x , then
f 1  x   log2  x  .
Notice that we had to write a subscript of 2 in the logarithm to indicate it “undoes” an exponential function with
a base of 2.
If 6 x  216 , then
log 6  6 x   log 6  216 
x  log 6  216 
x3
We know this is the correct answer because 63 = 216.
The logarithm to the base of 6 found the exponent, that when placed on a base of 6, gives an answer of 216.
The logarithm to the base of 6 is the inverse of the exponential function 6x: If f  x   6x , then
f 1  x   log6  x  .
Notice that we had to write a subscript of 6 in the logarithm to indicate it “undoes” an exponential function with
a base of 6.
If 10 x  100 , then
log 10 x   log 100 
x  log 100 
x2
We know this is the correct answer because 102 = 100.
The “common” logarithm found the exponent, that when placed on a base of 10, gives an answer of 100.
The “common” logarithm is the inverse of the exponential function 10x: If f  x   10x , then f 1  x   log  x  .
Notice that there is no sunscript written for a “common” logarithm; it is an agreed-upon notation that log(x)
“undoes” an exponential function with a base of 10.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 9.3
Page 3 of 8
Practice: Find the exact value of the following logarithmic expressions
1. log 2  32  (i.e., ask yourself, “what is the exponent on a base of 2 that gives an answer of 32?”)
2. log5 125 (i.e., ask yourself, “what is the exponent on a base of 5 that gives an answer of 125?”)
1
3. log 2  
8
1
4. log 3  
9
5. log 1 16 
2
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 9.3
Page 4 of 8
Converting Equations Between Exponential and Logarithmic Form
We have already practiced how to convert between power and root form of equations:
x 2  16

x  16
3
53  125

125  5
A similar kind of conversion between forms can be performed with exponential and logarithmic equations. Just
keep in mind that a logarithm is the exponent on the given base.
54  b

4  log5  b 
y  log3 81 
3 y  81
Practice: Convert the following equations between exponential and logarithmic form
6. 23  x
7. 4 n  16
8.
p 2  25
9. x  log2 16
1
10. 2  log a  
 4
11. 2  log5  y 
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 9.3
Page 5 of 8
Graphing Logarithmic Functions
To graph logarithmic functions, we will compute a table of points, then connect the points with a smooth curve.
You will notice that all logarithmic function graphs look kind of the same, just as the exponential function
graphs did.
Practice:
12. Graph the logarithmic function f  x   log2  x 
x
y  f  x   log2  x 
(x, y)
1/8
1/4
1/2
1
2
4
8
13. Graph the exponential function f  x   log3  x 
x
y  f  x   log3  x 
(x, y)
1/9
1/3
1
3
9
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 9.3
Page 6 of 8
Properties of the Graph of an Logarithmic Function, f  x   loga  x  , when a > 1






The domain is the set of all positive real numbers.
The range is the set of all real numbers.
There are no y-intercepts.
The x-intercept is 1.
1

The graph of f contains the points  , 1 , 1.0  , and  a,1 .
a

The function is always increasing.
Properties of the Graph of an Logarithmic Function, f  x   loga  x  , when 0 < a < 1






The domain is the set of all positive real numbers.
The range is the set of all real numbers.
There are no y-intercepts.
The x-intercept is 1.
1

The graph of f contains the points  , 1 , 1.0  , and  a,1 .
a

The function is always decreasing.
Natural and Common Logarithms
Definition: Common Logarithm
The common logarithm y  log  x  is defined and y  log10  x 
Definition: Natural Logarithm
The natural logarithm y  ln  x  is defined and y  loge  x 
Practice: Use your calculator to evaluate the following
14. log  25 
15. log  0.00195 
16. ln  25 
17. ln  0.00195 
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 9.3
Page 7 of 8
Solving Logarithmic Equations
 Re-write the logarithmic equation in exponential form, then use your knowledge of exponents and the fact
that if a u  a v , then u = v.
Practice: Solve the following logarithmic equations.
18. log3  4 x  7   2
19. log x  64   2
20. log5  25  x
21. ln  x   4
22. log  2 x  3  1
23. log7  2 x  7   log7  x  10
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 9.3
Page 8 of 8
Application of Logarithmic Functions: Loudness of a Sound
The loudness L, measured in decibels, of a sound of intensity x (measured in Watts per square meter) is given
by
 x 
L  x   10 log  12  .
 10 
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.