Exponential and Logarithmic Functions Questions

Section 6.4 Logarithmic Functions 183
Chapter 6: Exponential and Logarithmic Functions
Section 6.4: Logarithmic Functions
Exploration 1: Logarithms
Before we define a logarithm, let’s play around with them a little. See if you can follow the
pattern below to be able to fill in the missing pieces to a – f.
1
2
log 3 9 = 2
log 9 3 =
log 4 16 = 2
log 3 27 = 3
(a) log 2 8 = ___
(b) log 4 16 = ___
(c) log ___ 64 = 2
(d) log ___ 64 = 3
(e) log 2 ____ = 4
(f) log 4 2 = ___
Logarithms – A logarithm is just a power
For example, log 2 (32) = 5 says “the logarithm with base 2 of 32 is 5.” It means 2 to the 5th
power is 32. Notice that both in logarithms and exponents, the same number is called the
base.
The logarithmic function with base a, where a > 0 and a ≠ 1 , is denoted by y = log a x
(read as “y is the logarithm to the base a of x”) and is defined by:
________________________________
The domain of the logarithmic function y = logax is ___________.
Example 1*: Convert Exponential to Logarithmic Statements
Change each exponential equation to an equivalent equation involving a logarithm
(a) 58 = t
(b) x −2 = 12
(c) e x = 10
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184
Chapter 6 Exponential and Logarithmic Functions
Example 2*: Convert Logarithmic to Exponential Statements
Change each logarithmic equation to an equivalent equation involving an exponent.
(a) y = log 2 21
(b) log z 12 = 6
(c) log 2 10 = a
Example 3*: Evaluate Logarithmic Expressions
Evaluate the following:
1
(a)* log 3 (81)
(c) log 5 (1)
(b)* log 2
8
(e) log 3 (9)
(f) log 4 (2)
Let’s recall the domain and range
of an exponential function:
Range
Domain
f ( x) = a x
All Real
Numbers
(g) log1/3 (27)
(d) log 2 (16)
(h) log 5 (25)
Since a logarithmic function is the inverse of an
exponential function, fill in the domain and range
below based on what we learned in Section 6.2.
Domain
Range
f ( x) = log a x
All Real
Numbers
greater
than 0
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Section 6.4 Logarithmic Functions 185
Domain and Range of the Logarithmic Function y = log a ( x ) (defining equation x = a y )
Domain:__________________
Range:__________________
Example 4*: Determine the Domain of a Logarithmic Function
Find the domain of each logarithmic function.
 x+3
(a) f ( x ) = log3 ( x − 2 )
(b) F ( x ) = log 2 

 x −1 
(c) h ( x ) = log 2 x − 1
(d) g ( x ) = log 1 x 2
2
1.
2.
3.
4.
5.
6.
Properties of the Logarithmic Function f ( x) = log a ( x)
The domain _______________; The range is _______________.
The x-intercept is _______________. There is _______________ y-intercept.
The y-axis ( x = 0 ) is a ____________________ asymptote of the graph.
A logarithmic function is decreasing if __________ and increasing if __________.
The graph of f contains the points ___________________________.
The graph is _______________________________, with no _________________.
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186 Chapter 6 Exponential and Logarithmic Functions
Fact
Natural Logarithm: ln ( x ) means log e ( x ) . It is derived from the Latin phrase,
logarithmus naturalis. In other words, y = ln( x ) if and only if x = e y .
Example 5*: Graph Logarithmic Functions
(a)* Graph f ( x ) = 3ln( x − 1) .
(b)* State the domain of f ( x ) .
(c)* From the graph, determine the range and vertical asymptote of f.
(d) Find f −1 , the inverse of f.
(e) Use f −1 to confirm the range of f found in part (c). From the domain of f, find the range
of f −1 .
(f) Graph f −1 on the same set of axis as f.
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Section 6.4 Logarithmic Functions 187
Fact
Common Logarithm: log ( x ) means log10 ( x ) . In other words, y = log( x ) if and only if
x = 10 y .
Example 6: Graph a Logarithmic Functions
(a) Graph f ( x ) = −2 log ( x + 2 ) .
(b) State the domain of f ( x ) .
(c) From the graph, determine the range and vertical asymptote of f.
(d) Find f −1 , the inverse of f.
(e) Use f −1 to confirm the range of f found in part (c). From the domain of f, find the range
of f −1
(f) Graph f −1 on the same set of axis as f.
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188 Chapter 6 Exponential and Logarithmic Functions
Solving Basic Logarithmic Equations
When solving simple logarithmic equations (they will get more complicated in Section
4.6) follow these steps:
1. Isolate the logarithm if possible.
2. Change the logarithm to exponential form and use the strategies learned in Section
4.3 to solve for the unknown variable.
Example 7*: Solve Logarithmic Equations
Solve the following logarithmic equations
(b)* log x 343 = 3
(a)* log 2 ( 2x +1) = 3
(c) 6 − log
(d) ln ( x ) = 2
(f) log 6 36 = 5 x + 3
(e) 7 log 6 (4 x ) + 5 = −2
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=3
Section 6.4 Logarithmic Functions 189
Steps for solving exponential equations of base e or base 10
1. Isolate the exponential part
2. Change the exponent into a logarithm.
3. Use either the “log” key (if log base 10) or the “ln” (if log base e) key to evaluate the
variable.
Example 8*: Using Logarithms to Solve Exponential Equations
Solve each exponential equation.
(a) e x = 7
(b)* 2e3 x = 6
(c) e5 x −1 = 9
(d) 4(102 x ) + 1 = 21
(e) 3e 2 x +1 − 2 = 10
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190
Chapter 6 Exponential and Logarithmic Functions
Chapter 6: Exponential and Logarithmic Functions
Section 6.5: Properties of Logarithms
Exploration 1: Establish Properties of Logarithms
Calculate the following:
(a) log 5 (1)
(b) log 2 (1)
(c) log(1)
(d) ln(1)
(e) log 5 (5)
(h) ln(e)
(f) log 2 (2)
(g) log(10)
Properties of Logarithms:
To summarize:
1. log a 1 = _______
2. log a a = _______
Exploration 2: Establish Properties of Logarithms
In section 6.4, we found that the inverse of the function f ( x ) = log 2 ( x ) was f −1 ( x ) = 2 x . In
fact, in general we can say that the functions defined by g ( x ) = log a ( x) and h( x) = a x are
inverse functions. Knowing what you know about inverse functions, evaluate:
(a) g (h(r ))
(b) h( g (m))
Properties of Logarithms:
To summarize: In the following properties, M and a are positive real numbers, where a ≠ 1 ,
and r is any real number :
3. log a a r = _______
4. a log a M = _______
Exploration 3: Establish Properties of Logarithms
Show that the following are true
 1000 
(a) log (100 ⋅ 10 ) = log(100) + log(10) (b) log 
 = log(1000) − log(100)
 100 
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(c) log103 = 3 log(10)
Section 6.5 Properties of Logarithmic 191
Properties of Logarithms:
To summarize: In the following properties, M, N, and a are positive real numbers, where
a ≠ 1 , and r is any real number :
M
5. log a ( MN ) = __________ 6. log a 
N

r
 = __________ 7. log a M = ________

Example 1*: Work with the Properties of Logarithms
Use the laws of logarithms to simplify the following:
20
(a) 3log3 18
(b) 2log 2 ( −5)
1
(c) log 1  
 
2 2
(d) ln(e3 )
Example 2: Work with the Properties of Logarithms
Use the laws of logarithms to find the exact value without a calculator.
(b) log 8 (2) − log 8 (32)
(a) log 3 (24) − log 3 (8)
(c) 6log6 (3) + log6 (5)
(d) e
log
e2
(25)
Example 3*: Write a Logarithmic Expression as a Sum or Difference of Logarithms
Write each expression as a as a sum or difference of logarithms. Express all powers as
factors.
 x2 y3 
2
(a) log 3 ( x − 1)( x + 2 )  , x > 1
(b) log 5 



 z 
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192 Chapter 6 Exponential and Logarithmic Functions
Example 4*: Write a Logarithmic Expression as a Single Logarithm
Write each of the following as a single logarithm.
(b) 3log 6 z − 2 log 6 y
(a) log 2 x + log 2 ( x − 3 )
1
(c) ln ( x − 2 ) + ln x − 5ln ( x + 3)
2
Properties of Logarithms continued:
In the following properties, M, N, and a are positive real numbers where a ≠ 1 :
8. If M = N, then ___________________
9. If log a M = log a N , then ___________
Let a ≠ 1, and b ≠ 1 be positive real numbers. Then the change of base formula says:
10. log a M = _____________
Why would we want to use the change of base formula?
Example 5*: Evaluate a Logarithm Whose Base is Neither 10 nor e.
Approximate the following. Round your answers to four decimal places.
(b) log 7 325
(a) log 3 12
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Section 6.5 Properties of Logarithmic 193
Summary Properties of Logarithms:
In the following properties, M, N, and a are positive real numbers, where a ≠ 1 , and r is any
real number :
log a 1 = _______
log a a = _______
log a M r = _______
a log a M = _______
log a a r = _______
a r = _______
M
log a 
N
If M = N, then ___________________
If
log a ( MN ) = ______________

 = ______________

log a M = log a N , then ___________
Change of base formula: log a M = _____________
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