Properties of Logarithms; Logarithmic and Exponential Equations
From last week (5.4)
Post 1
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Author in Action: Solve Logarithmic Equations (6:43)
Solving Basic Logarithmic Equations
When solving simple logarithmic equations (they will get more complicated in Section 5.6) follow
these steps:
1. Isolate the logarithm if possible.
2. Change the logarithm to exponential form and use the strategies learned in Section 5.3 to solve
for the unknown variable.
Example 7*: Solve Logarithmic Equations
Solve the following logarithmic equations
(a)* log2 ( 2x +1) = 3
(b)* log x 343 = 3
End of Post 1
(c) 6 − log(𝑥) = 3
(d) ln ( x ) = 2
(e) 7 log 6 (4 x) + 5 = −2
(f) log 6 36 = 5 x + 3
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Section 5.5 & 5.6
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
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Properties of Logarithms; Logarithmic and Exponential Equations
Section 5.5: Properties of Logarithms
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Author in Action: Work with Properties of Logarithms (12:32)
Exploration 1: Establish Properties of Logarithms
Calculate the following:
(a) log5 (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 5.4, we found that the inverse of the function f ( x) = log 2 ( x) was f −1 ( x) = 2x . 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
(c) log103 = 3log(10)
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Section 5.5 & 5.6
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. loga ( MN ) = __________ 6. log a
N
r
= __________ 7. log a M = ________
Post 2
Example 1*: Work with the Properties of Logarithms
Use the laws of logarithms to simplify the following:
æ1ö
(c) log 1 ç ÷
è ø
2 2
20
(a) 3log3 18
(b) 2log2 ( −5)
End of Post 2
Example 2: Work with the Properties of Logarithms
Use the laws of logarithms to find the exact value without a calculator.
(a) log3 (24) − log3 (8)
(b) log8 (2) − log8 (32)
(c) 6log6 (3) + log6 (5)
(d) e
log
e2
(25)
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(d) ln(e3 )
Properties of Logarithms; Logarithmic and Exponential Equations
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Author in Action: Write a Logarithmic Expression as a Sum or Difference of Logarithms (9:47)
Post 3
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) log3 ( x − 1)( x + 2 ) , x 1
(b) log5
z
End of Post 3
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Author in Action: Write a Logarithmic Expression as a Single Logarithm (5:47)
Post 4
Example 4*: Write a Logarithmic Expression as a Single Logarithm
Write each of the following as a single logarithm.
(a) log2 x + log2 ( x − 3)
(b) 3log6 z − 2log6 y
1
(c) ln ( x − 2 ) + ln x − 5ln ( x + 3)
2
End of Post 4
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Author in Action: Evaluate a Logarithm Whose Base Is Neither 10 Nor e (9:20)
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 = _____________
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Section 5.5 & 5.6
Why would we want to use the change of base formula?
Post 5
Example 5*: Evaluate a Logarithm Whose Base is Neither 10 nor e.
Approximate the following. Round your answers to four decimal places.
(a) log3 12
(b) log7 325
End of Post 5
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 loga M = _______
log a a r = _______
a r = _______
M
log a
N
If M = N, then ___________________
If
loga ( MN ) = ______________
= ______________
log a M = log a N , then ___________
Change of base formula: log a M = _____________
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Properties of Logarithms; Logarithmic and Exponential Equations
Section 5.6: Logarithmic and Exponential Equations
We will use the properties of logarithms found in Section 5.5 to solve all types of equations where a
variable is an exponent. The following definition and properties that we’ve seen in previous sections will
be particularly useful and provided here for your review:
Summary: Know this!
The logarithmic function to the 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:
y = log a x if and only if x = a y
The domain of the logarithmic function y = logax is x > 0.
Properties of Logarithms:
In the following properties, M, N, and a are positive real numbers, where a 1 , and r is any real
number :
1. a log a M = M
2. log a a r = r
3. loga ( MN ) = log a M + log a N
M
4. log a
= log a M − log a N
N
5. log a M r = r log a M
6. a x = e x ln a
7. If M = N , then loga M = loga N
8. If loga M = loga N , then M = N .
Strategy for Solving Logarithmic Equations Algebraically
1. Rewrite the equation using properties of logarithms so that it is written in one of the following two
ways: log a x = c or log a (something) = log a (something else) .
2. If the equation is of the form log a x = c change it to exponential form to undo the logarithm and
solve for x.
3. If the equation is of the form log a (something) = log a (something else) use property 8 to get rid of
the logarithms and solve.
4. Check your solutions. Remember that The domain of the logarithmic function y = logax is x > 0.
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Section 5.5 & 5.6
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Properties of Logarithms; Logarithmic and Exponential Equations
Example 1: Solve Logarithmic Equations
Solve the following equations:
(a) log3 4 = 2log3 x
(b) log2 ( x + 2) + log2 (1 − x ) = 1
Example 2: Solve Logarithmic Equations
Solve the following equations:
(a) ln ( x −1) + ln x = ln ( x + 2)
(c) log(1 − c) = 1 + log(1 + c)
(b) log 4 (h + 3) − log 4 (2 − h) = 1
(d) ln(3m + 1) = 2 + ln(m − 3)
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Section 5.5 & 5.6
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Author in Action: Solve Exponential Equations (20:52)
Example 3*: Solve Exponential Equations
Solve the following equations:
(a) 9 x − 3x − 6 = 0
(b) 3 x = 7
(c) 5 2 x = 3
(d) 2 x −1 = 52 x +3
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Properties of Logarithms; Logarithmic and Exponential Equations
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Solve Logarithmic and Exponential Equations Using a Graphing Utility (2:32)
So far we have solved exponential and logarithmic equations algebraically. Another method we can use
is to solve by graphing. Here is a list of steps for how to do this:
Solving by Graphing
1.
2.
3.
4.
Put one side of the equation in y1 .
Put one side of the equation in y2 .
Graph the equations and find the point at which they intersect.
The x value is your solution.
Example 4*: Solving Logarithmic and Exponential Equations Using a Graphing Utility
Solve e x = − x using a graphing utility.
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