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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

Copyright © 2016 Pearson Education, Inc.

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

Copyright © 2016 Pearson Education, Inc.

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)

Copyright © 2016 Pearson Education, Inc.

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)

Copyright © 2016 Pearson Education, Inc.

(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 = _____________

Copyright © 2016 Pearson Education, Inc.

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 = _____________

Copyright © 2016 Pearson Education, Inc.

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.

Copyright © 2016 Pearson Education, Inc.

Section 5.5 & 5.6

Copyright © 2016 Pearson Education, Inc.

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)

Copyright © 2016 Pearson Education, Inc.

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

Copyright © 2016 Pearson Education, Inc.

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.

Copyright © 2016 Pearson Education, Inc.

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