Logarithmic Functions Objectives Recognize and evaluate logarithmic functions with base a. Graph...
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Transcript of Logarithmic Functions Objectives Recognize and evaluate logarithmic functions with base a. Graph...
Logarithmic Functions
Objectives
Recognize and evaluate logarithmic functions with base a.
Graph logarithmic functions with base a.
Recognize, evaluate, and graph logarithmic functions with base e.
Definition: Logarithmic Function
For x 0 and 0 a 1, y = loga x if and only if x = a y.
The function given by f (x) = loga x is called the
logarithmic function with base a.
Every logarithmic equation has an equivalent exponential form: y = loga x is equivalent to x = a y
A logarithmic function is the inverse function of an exponential function.
Exponential function: y = ax
Logarithmic function: y = logax is equivalent to x = ay
A logarithm is an exponent!
y = logax if and only if x = a y
The logarithmic function to the base a, where a > 0 and a 1 is defined:
2416
exponential formlogarithmic form
Convert to log form: 216log4 Convert to exponential form:
38
1log2
8
12 3
When you convert an exponential to log form, notice that the exponent in the exponential becomes what the log is equal to.
LOGS = EXPONENTS
With this in mind, we can answer questions about the log:
16log2
This is asking for an exponent. What exponent do you put on the base of 2 to get 16? (2 to the what is 16?)4
9
1log3
What exponent do you put on the base of 3 to get 1/9? (hint: think negative)2
1log4
What exponent do you put on the base of 4 to get 1?
0
3log3
When working with logs, re-write any radicals as rational exponents.
2
1
3 3logWhat exponent do you put on the base of 3 to get 3 to the 1/2? (hint: think rational)2
1
Examples: Write Equivalent Equations
y = log2( )2
1= 2
y
2
1
Examples: Write the equivalent exponential equation and solve for y.
1 = 5 yy = log51
16 = 4y y = log416
16 = 2yy = log216
SolutionEquivalent Exponential
Equation
Logarithmic Equation
16 = 24 y = 4
2
1= 2-1 y = –1
16 = 42 y = 2
1 = 50 y = 0
Your Turn:• Write each equation in exponential form
log 3 81 = 434=81
log 7 1/49 = -27-2=1/49
• Write each equation in logarithmic form103 = 1000
Log101000=34-2 = 1/16
Log41/16=-2
Properties of Logarithms
Examples: Solve for x: log6 6 = x
log6 6 = 1 property 2 x = 1
Simplify: log3 35
log3 35 = 5 property 3
Simplify: 7log7
9
7log7
9 = 9 property 3
Properties of Logarithms
1. loga 1 = 0 since a0 = 1.
2. loga a = 1 since a1 = a.
4. If loga x = loga y, then x = y. one-to-one property
3. loga ax = x and alogax = x inverse property
Base 10 logarithms
• Called common logarithms• When base a is not indicated, it is
understood that a = 10• log 1/100 = log 10 = • log 1/10 = log 100 =• log 1 = log 1000 =• The LOG key on your calculator.
-2
-1
0
1
2
3
In the last section we learned about the graphs of exponentials.
Logs and exponentials are inverse functions of each other so let’s see what we can tell about the graphs of logs based on what we learned about the graphs of exponentials.
Recall that for functions and their inverses, x’s and y’s trade places. So anything that was true about x’s or the domain of a function, will be true about y’s or the range of the inverse function and vice versa.
Let’s look at the characteristics of the graphs of exponentials then and see what this tells us about the graphs of their inverse functions which are logarithms.
Characteristics about the Graph of an Exponential Function a > 1
xaxf 1. Domain is all real numbers
2. Range is positive real numbers
3. There are no x intercepts because there is no x value that you can put in the function to make it = 0
4. The y intercept is always (0,1) because a 0 = 1
5. The graph is always increasing
6. The x-axis (where y = 0) is a horizontal asymptote for x -
Characteristics about the Graph of a Log Function a > 1
xxf alog1. Range is all real numbers
2. Domain is positive real numbers
3. There are no y intercepts
4. The x intercept is always (1,0) (x’s and y’s trade places)
5. The graph is always increasing
6. The y-axis (where x = 0) is a vertical asymptote
Exponential Graph Logarithmic Graph
Graphs of inverse functions are reflected about the line y = x
basea>1
Graph f(x) = log2 x
x
y
Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x.
83
42
21
10
–1
–2
2xx
4
1
2
1
y = log2 x
y = xy = 2x
(1, 0)
x-intercept
horizontal asymptote y = 0
vertical asymptote x = 0
Graphs of Logarithmic Functions
The graphs of logarithmic functions are similar for different values of a. f(x) = loga x (a 1)
3. x-intercept (1, 0)
5. increasing
6. continuous
7. one-to-one
8. reflection of y = a x in y = x
1. domain ),0( 2. range ),(
4. vertical asymptote
)(0 as 0 xfxx
Graph of f (x) = loga x (a 1)
x
yy = x
y = log2 x
y = a x
domain
range
y-axisverticalasymptote
x-intercept(1, 0)
Graphs of Logarithmic Functions
• Typical shape for graphs where a > 1 (includes base e and base 10 graphs).
• Typical shape for graphs where 0 < a < 1.
Determining Domains of Logarithmic Functions
Example Find the domain of each function.
Solution
(a) Argument of the logarithm must be positive.
x – 1 > 0, or x > 1. The domain is (1,).(b) Use the sign graph to solve x2 – 4 > 0.
)4ln()( (b) )1(log)( (a) 22 xxfxxf
The domain is (–,–2) (2, ).
Your Turn:
Find the domain.
(a)
Solutiona) Domain: (-3,) or x >-3b) Domain: (-3, 3) or -3<x <3
22 3log ( 3) (b) log 9y x y x
• Any log to the base e is known as a
natural logarithm.• In French this is a
logarithme naturel• Which is where ln comes from.• When you see ln (instead of log)
– then it’s a natural log• y = ln x is the inverse of y = ex
• The LN key on your calculator.
Natural Logarithms