In this section we will be working with Properties of Logarithms in an ...

16-week Lesson 33 (8-week Lesson 27)

Properties of Logarithms and Solving Log Equations (Part 2)

In this section we will be working with Properties of Logarithms in an

attempt to take equations with more than one logarithm and condense

them down into just a single logarithm.

Properties of Logarithms:

a. Product Rule:

log ? (?) + log ? (?) = log ? (? ? ?)

When two or more logarithms with the same base are added, those

logarithms can be condensed into one logarithm whose argument is

the product of the original arguments (? ? ? in the example above)

Order is not important when multiplying or adding, so changing the order

of the factors in an argument or changing the order of the terms being

added together does not change the answer.

b. Quotient Rule:

?

log ? (?) ? log ? (?) = log ? ( )

?

When two or more logarithms with the same base are subtracted,

those logarithms can be condensed into one logarithm whose

argument is the quotient of the original arguments

?

(? ?? ??? ??????? ?????)

Order is important when dividing and subtracting; the numerator is always

the argument of the first log term listed (or the argument of the term listed

first is always the numerator).

c. Power Rule:

? ? log ? (?) = log ? (?? )

A factor times a logarithm can be re-written as the argument of the

logarithm raised to the power of that factor

1

16-week Lesson 33 (8-week Lesson 27)

Properties of Logarithms and Solving Log Equations (Part 2)

Example 1: Solve each of the following logarithmic equations and

CHECK YOUR SOLUTIONS. LEAVE ANSWERS IN EXACT

FORM, DO NOT APPROXIMATE.

a. 2 = log 3 (?) ? log 3 (? ? 1)

b. A

?

2 = log 3 (??1)

32 =

b. log 2 (? + 7) + log 2 (? ) = 3

log 2 ((? + 7)(? )) = 3

?

log 2 (? 2 + 7? ) = 3

??1

?

9 = ??1

? 2 + 7? = 23

9(? ? 1) = ?

? 2 + 7? = 8

9? ? 9 = ?

? 2 + 7? ? 8 = 0

8? = 9

(? + 8)(? ? 1) = 0

9

?=8

? = ?8 ; ? = 1

Regardless of what type of answer you come up with (negative,

positive, or zero), you must check your answer to verify that it results

in a positive argument. To do so, ALWAYS plug your answer back

into the original equation.

9

Replacing ? with 8 will make

each argument in the original

9

equation positive, so ? = 8

is a valid answer.

?=

Replacing ? with ?8 will make

each argument in the original

equation negative, so ? = ?8

is not a valid answer. ? = 1 is a

valid answer because it makes

the arguments positive.

?

?=?

?

2

16-week Lesson 33 (8-week Lesson 27)

Properties of Logarithms and Solving Log Equations (Part 2)

In Example 1, the Properties of Logarithms were only used to combine

logarithms in each problem. This is how we will be using the Properties

of Logarithms in this class, to combine logarithms in order to reduce the

number logarithms we have to just one, so that we can then convert that

one logarithm to exponential form to solve.

If an equation has more than one logarithm on either side of the equation,

use the Properties of Logarithms to simplify as much as possible, then

solve by converting to exponential form. Remember that you must

check your answers when solving log equations to verify that they

make the original arguments positive.

Example 2: Solve each of the following logarithmic equations and

CHECK YOUR SOLUTIONS. LEAVE ANSWERS IN EXACT

FORM, DO NOT APPROXIMATE.

a. 2 ln(? ) ? ln(5? ? 6) = 0

b. log 2 (1 ? ? ) = 1 ? log 2 (? ? 1)

?

2 = log 3 (??1)

log 2 (1 ? ? ) + log 2 (? ? 1) = 1

?

log 2 ((1 ? ? )(? ? 1)) = 1

2 = log 3 (??1)

?

log 2 (? ? 1 ? ? 2 + ? ) = 1

2 = log 3 (??1)

?

log 2 (?? 2 + 2? ? 1) = 1

2 = log 3 (??1)

?

?? 2 + 2? ? 1 = 21

2 = log 3 (??1)

?

?? 2 + 2? ? 1 = 2

2 = log 3 (??1)

?

2 = log 3 (??1)

?? + 2? ? 3 = 0

2 = log 3 (??1)

?

? 2 ? 2? + 3 = 0

completing

2

?2 2

?

The

quadratic

equation I

ended up

with was

not

factorable,

so I chose

to solve it

by

?2

the square.

When I

did, I

ended up

with an

equation

2 that is not

possible

using real

numbers.

2 = log 3 (??1)

? 2 ? 2? + ( ) = ?3 + ( )

2 = log 3 (??1)

? 2 ? 2? + 1 = ?3 + 1

2 = log 3 (??1)

(? ? 1)2 = ?2

2

?

?

?? ????????

3

2

Therefore

the log

equation

given on

part b. has

no real

solutions.

16-week Lesson 33 (8-week Lesson 27)

Properties of Logarithms and Solving Log Equations (Part 2)

b.

1

c. 2 log(? + 1) = log(1 ? ? )

a

This example demonstrates

using two different Properties

of Logarithms (the Power Rule

first and the Quotient Rule

second). Notice that when

log(¡Ì? + 1) = log(1 ? ?), we

could have simply set the

arguments equal to each other

and solved. In other words, if

log(¡Ì? + 1) = log(1 ? ?),

then ¡Ì? + 1 = 1 ? ?. You can

see in the solution that we end

up with this equation

eventually, but only after using

the Quotient Rule for

Logarithms and converting

from log form to exponential

form. This shortcut can be used

because logarithmic functions

are one-to-one functions, so if

log(?) = log(?), then ? = ?.

1

2

log(? + 1) = log(1 ? ? )

log(¡Ì? + 1) = log(1 ? ? )

log(¡Ì? + 1) ? log(1 ? ? ) = 0

¡Ì?+1

log ( 1?? ) = 0

¡Ì?+1

1??

= 100

¡Ì?+1

1??

=1

¡Ì? + 1 = 1 ? ?

2

(¡Ì? + 1) = (1 ? ? )2

You are welcome to use this

shortcut or not, it makes no

difference; your answers should

be the same regardless. If you

plan to use this shortcut, please

keep in mind this only works

when you have one logarithm

equal to another. You cannot

take this shortcut if you have

any other terms or factors in the

equation (such as Example 2

parts a. and b. on the previous

page).

? + 1 = (1 ? ? )(1 ? ? )

? + 1 = 1 ? 2? + ? 2

0 = ? 2 ? 3?

0 = ? (? ? 3)

0=? ; ??3=0

?=0 ; ?=3

Replacing ? with 0 in the original equation makes each argument

positive, so ? = 0 is a valid answer. Replacing ? with 3 results in a

negative argument (1 ? 3 = ?2), so ? = 3 is not a valid answer.

?=?

4

16-week Lesson 33 (8-week Lesson 27)

Properties of Logarithms and Solving Log Equations (Part 2)

Example 3: Solve each of the following logarithmic equations and

CHECK YOUR SOLUTIONS. LEAVE ANSWERS IN EXACT

FORM, DO NOT APPROXIMATE.

a. log(? + 2) ? log(?) = 2 log(4)

b. log 4 (3? + 2) = log 4 (5) + log 4 (3)

5

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