Download Section12.2

12.2 Logarithmic Functions
By the horizontal–line test, all exponential functions are one–to–one
functions. Therefore, they have an inverse function.
Let’s determine the inverse function of an exponential function. We do
this by interchanging the variables and solving for y. (Section 8.4)
𝑓(𝑥 ) = 𝑏 𝑥
𝑦 = 𝑏𝑥
𝑥 = 𝑏𝑦
𝑦 = the power to which we raise 𝑏 to get 𝑥
In order to write this relationship mathematically, we define the
logarithm of x base b to be the power to which we raise b to get x.
“logarithm of x base b” is written 𝒍𝒐𝒈𝒃 𝒙
𝒚 = 𝒍𝒐𝒈𝒃 𝒙
if and only if
𝒙 = 𝒃𝒚
where b and x are positive real numbers and 𝑏 ≠ 1
By definition, a logarithm is an exponent.
------------------------------------------------------------------------------Now we can write the inverse function of the exponential function,
𝑓(𝑥 ) = 𝑏 𝑥 as
𝑦 = 𝑙𝑜𝑔𝑏 𝑥
or
𝑓 −1 (𝑥 ) = 𝑙𝑜𝑔𝑏 𝑥
Examples: Write the inverse function for 𝑔(𝑥 ) = 2𝑥 .
𝑔−1 (𝑥 ) = 𝑙𝑜𝑔2 𝑥
Write the inverse function for 𝑓(𝑥 ) = 5𝑥 .
𝑓 −1 (𝑥 ) = 𝑙𝑜𝑔5 𝑥
Properties of Logarithms
For 𝑏 > 0 and 𝑏 ≠ 1,
𝑙𝑜𝑔𝑏 𝑏 = 1
𝑙𝑜𝑔𝑏 1 = 0
𝑙𝑜𝑔𝑏 𝑏 𝑥 = 𝑥
𝑏 𝑙𝑜𝑔𝑏 𝑥 = 𝑥
because 𝑏1 = 𝑏
because 𝑏 0 = 1
because 𝑏 𝑥 = 𝑏 𝑥
because log b 𝑥 is the exponent with base 𝑏
that equals 𝑥
𝑙𝑜𝑔𝑏 (−1) = undefined
𝑙𝑜𝑔𝑏 0 = undefined
because b is positive and no power
of b equals a negative number or zero
See Examples 4 and 5 on page 834.
-------------------------------------------------------------------------------------Common Logarithms
The base of a logarithm can be any positive real number except 1.
However, one base that appears more often than other rational numbers
is the base 10. A base–10 logarithm is called a common logarithm.
Base–10 logarithms are written 𝒍𝒐𝒈 𝒙. The base is not written.
On your calculator, the LOG key is used to evaluate common
logarithms. (See the table at the top of page 837.)
Examples: 𝑙𝑜𝑔 1 = 0
because 100 = 1
𝑙𝑜𝑔 0.01 = −2
because 10−2 =
1
100
= 0.01
𝑙𝑜𝑔 23 ≈ 1.3617 (use your calculator)
𝑙𝑜𝑔 0 = undefined
101.3617 ≈ 23
argument must be a positive real
NOTE: your calculator will show the error message ERR: DOMAIN
when the logarithm is undefined
Natural Logarithms
Logarithms with base e are called natural logarithms. Natural
logarithms are written with a special symbol ln x and is read “the natural
log of x” or “el en of x.” The base is not written.
All the properties of logarithms listed in Section 12.2 hold for natural
logarithms as well.
because 𝑒 0 = 1
because 𝑒 1 = 𝑒
because 𝑒 𝑚 = 𝑒 𝑚
because ln 𝑥 is the exponent with base 𝑒
that equals 𝑥
ln 1 = 0
ln 𝑒 = 1
𝑙𝑛 𝑒 𝑚 = 𝑚
𝑒 𝑙𝑛 𝑥 = 𝑥
𝑙𝑛 (– 1) = undefined
𝑙𝑛 0 = undefined
because e is positive and no power of
e equals a negative number or zero
On your calculator, the LN key is used to evaluate natural logarithms.
Examples: 𝑙𝑛 𝑒 3 = 3
𝑙𝑛
1
𝑒2
= −2
because 𝑒 3 = 𝑒 3
because 𝑒 −2 =
1
𝑒2
𝑙𝑛 8 ≈ 2.0794 because 𝑒 2.0794 ≈ 8
𝑙𝑛 (−3) = undefined
argument must be a positive real
NOTE: your calculator will show the error message
ERR: NONREAL ANS when the logarithm is undefined
--------------------------------------------------------------------------------------
Changing Equation Forms
The equations 𝑦 = 𝑙𝑜𝑔𝑏 𝑥 and 𝑥 = 𝑏 𝑦
are different ways of
expressing the same idea. The first equation is in logarithmic form and
the second equation is in exponential form.
To change an equation from one form to the other form
 identify the 3 parts of the given equation
 place each part in its appropriate position in the alternate form.
Exponential Equation: 32 = 9
In this equation, 3 is the BASE, 2 is the EXPONENT and 9 is the
VALUE.
Logarithmic Equation: 𝑙𝑜𝑔3 9 = 2
In this equation, 3 is the BASE, 9 is the VALUE (or ARGUMENT) and
2 is the EXPONENT.
Let B = base, E = exponent and V = value. The two equation forms are
Exponential Equation: 𝐵𝐸 = 𝑉
Logarithmic Equation: 𝑙𝑜𝑔𝐵 𝑉 = 𝐸
𝑙𝑜𝑔 𝑉 = 𝐸 Common (base–10) Logarithms
𝑙𝑛 𝑉 = 𝐸 Natural (base–e) Logarithms
Writing Exponential Equations as Logarithmic Equations
Write each exponential equation as a logarithmic equation.
1⁄
2
23 = 8
36
3−4 =
𝑙𝑜𝑔2 8 = 3
𝑙𝑜𝑔36 6 =
𝑙𝑜𝑔 10 = 1
𝑙𝑜𝑔 0.01 = −2
=6
1
𝑙𝑜𝑔3
1
81
1
= −4
2
81
-------------------------------------------------------------------------------------------------------------------1
101 = 10
10−2 = 0.01
10 ⁄4 ≈ 1.7783
𝑙𝑜𝑔 1.7783 ≈
1
4
𝑒 3 ≈ 20.0855
𝑒 2.0794 ≈ 8
𝑒 −1 ≈ 0.3679
𝑙𝑛 20.0855 ≈ 3
𝑙𝑛 8 ≈ 2.0794
𝑙𝑛 0.3679 ≈ −1
-------------------------------------------------------------------------------------Writing Logarithmic Equations as Exponential Equations
Write each logarithmic equation as an exponential equation.
𝑙𝑜𝑔2 32 = 5
𝑙𝑜𝑔8 2 =
25 = 32
8
1⁄
3
1
3
𝑙𝑜𝑔4
1
16
4−2 =
=2
= −2
1
16
-------------------------------------------------------------------------------------1
𝑙𝑜𝑔 100 = 2
𝑙𝑜𝑔
102 = 100
10−4 =
10,000
= −4
𝑙𝑜𝑔 0.001 = −3
1
10−3 = 0.001
10,000
-------------------------------------------------------------------------------------𝑙𝑛 12 ≈ 2.4849
𝑙𝑛 0.0498 ≈ −3
𝑙𝑛 1.6487 ≈
𝑒 2.4849 ≈ 12
𝑒 −3 ≈ 0.0498
𝑒
1⁄
2
1
2
≈ 1.6487
---------------------------------------------------------------------------------------------Evaluating Logarithms
Remember: a logarithm is an exponent
When asked to find 𝑙𝑜𝑔𝑏 𝑥 ask the question: “x is equal to what power
of b?” The power (exponent) is the logarithm.
Examples: Evaluate 𝑙𝑜𝑔2 32.
32 = 2? ? = 5
𝑙𝑜𝑔2 32 = 𝑙𝑜𝑔2 25 = 5.
--------------------------------------------------------------------------------
Evaluate 𝑙𝑜𝑔4 2.
2 = 4? (2 = √4 = 4
1⁄
2)
? = 1⁄2
1
𝑙𝑜𝑔4 2 = 𝑙𝑜𝑔4 41⁄2 = .
2
-------------------------------------------------------------------------------------1
Evaluate 𝑙𝑜𝑔3 .
9
1
9
1
1
9
32
= 3? ( =
𝑙𝑜𝑔3
1
9
= 3−2 )
? = −2
= 𝑙𝑜𝑔3 3−2 = −2.
-------------------------------------------------------------------------------Graphing Logarithmic Functions
There are two methods for graphing a logarithmic function.
METHOD 1:
 set up a table of values for the inverse of the logarithmic function –
an exponential function
 reverse the coordinates to determine a table of values for the
logarithmic function
 plot the points
 connect the points with a smooth, continuous curve
 put arrowheads at both ends of the graph
METHOD 2:
 set up a table of values for the logarithmic function
 plot the points from the table
 connect the points with a smooth, continuous curve
 put arrowheads at both ends of the graph
Example: Graph 𝑓 (𝑥 ) = 𝑙𝑜𝑔2 𝑥.
METHOD 1:
 set up a table of values for the inverse function, 𝑓 −1 (𝑥 ) = 2𝑥
𝑥
𝑦 = 2𝑥
−3 2−3 = 1⁄
8
−2
1
−2 2 = ⁄
4
−1
1
−1 2 = ⁄
2
0
0 2 =1
1 21 = 2
2 22 = 4
3 23 = 8
 reverse the coordinates to determine a table of values for the
logarithmic function, 𝑓 (𝑥 ) = 𝑙𝑜𝑔2 𝑥
𝑥 𝑦 = 𝑙𝑜𝑔2 𝑥
1⁄
−3
8
1⁄
−2
4
1⁄
−1
2
1
0
2
1
4
2
8
3
 plot the points from this table
 connect the points with a smooth, continuous curve
 put arrowheads on both ends of the graph
y
10
5
f(x) = log2 x
-10
-5
5
10
x
-5
-10
METHOD 2:
 set up a table of values for the logarithmic function
Use values for x that are integral powers of 2.
𝑥 𝑦 = 𝑙𝑜𝑔2 𝑥
1⁄
−3
8
1⁄
−2
4
1⁄
−1
2
1
0
2
1
4
2
8
3
 plot the points from the table
 connect the points with a smooth, continuous curve
 put arrowheads on both ends of the graph
The graph is the same as with METHOD 1
Characteristics of a Logarithmic Function and Its Graph
For the logarithmic function, 𝑓(𝑥 ) = 𝑙𝑜𝑔𝑏 𝑥, and its graph,
 Domain: the set of all positive real numbers {𝑥 | 𝑥 > 0}
 Range: the set of all real numbers
(𝟎, ∞)
{𝒙| 𝒙 𝐢𝐬 𝐚 𝐫𝐞𝐚𝐥 𝐧𝐮𝐦𝐛𝐞𝐫} (−∞, ∞)
 The function is one-to-one
 y–intercept: there is no y–intercept.
The y–axis is a vertical asymptote.
 x–intercept: (1,0)
 𝑏 > 1 the function is increasing
0 < 𝑏 < 1 the function is decreasing
-------------------------------------------------------------------------------------Modeling the Real World
The values of a logarithmic function change slowly over a large range of
values. Thus, logarithmic functions model real-world situations where
the independent variable takes on a wide range of values.
Magnitude of an Earthquake
Measured on the Richter Scale: 𝑹 = 𝐥𝐨𝐠
𝑰
𝑰𝟎
𝑰𝟎 is the intensity of a very small vibration in the Earth used as a
standard.
Loudness of a Sound
Measured in decibels: 𝑫 = 𝟏𝟎𝒍𝒐𝒈(𝟏𝟎𝟏𝟐 𝑰)
I is the intensity of the sound, in watts per meter2.
pH Value of a Substance
Measures the degree of acidity or alkalinity: 𝒑𝑯 = − 𝐥𝐨𝐠 [𝑯+ ]
𝑯+ is the hydrogen ion concentration in moles per liter.
Pure water has a pH value of 7.0
greater than 7.0 is an alkaline substance
less than 7.0 is an acidic substance