Solving exponential and logarithmic equations

1
Solving exponential and logarithmic equations

• Section 5.5

2
Some thoughts on and its inverse

• If a function is one-to-one then for every y
  value there is only one x value. So if for
  some numbers u and v then .

Lets say Jack is thinking of a number whose log
base 10 is 2 and Jill is thinking of a number
whose log base 10 is 2. Since there is only one
such number you know they are thinking of the
same number.
3

• If a relationship is a function then for every x
  value there is only one y value. So if
  for some numbers u and v then .

This means that we cannot take the log base 10
of 100 and get two different answers. There is
just one number x such that
4
All of these functions are 1-1.
5
Rules

• Since is 1-1 we can say that if
  then
• . (Here u and v are real numbers and a is
  a positive real number not 1.)
• Since is 1-1 we can say that if
  then . (Here M N and a are positive real
  numbers and a is not 1.)

If then
If then
6
Rules
Since is a function we can say that if
then . (Here u and v are real numbers
and a is a positive real number not 1.) Since
is a function we can say that if
then . (Here M N and a are
positive real numbers and a is not 1.)
If then
If then
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• Remember and undo each other. They are
  inverses and so whatever the one does the other
  will undo. The last two rules will make use of
  this inverse relationship.
• In 5.5 were solving equations. Remember we are
  trying to find the x that makes the equation
  true. We will need to keep a lot in our minds.
• 1.) the four rules just discussed
• 2.) the fact that and are inverses
• 3.) the fact that and are equivalent
• 4.) the logarithm rules from 5.4

8
expl

• Solve
• Two different ways
• Method 1

Use equivalency of and
and simplify.
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Use if then and the inverse relationship

• Method 2

The logarithm function undid the exponential
function and left 4w on the left.
Apply to both sides.
10
expl
Use equivalency of and

• Solve
• Two different ways
• Method 1

Use the Change of Base formula on left
and simplify.
Check
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If then

• Method 2

take log of both sides
and simplify.
12
expl 9

• Solve

Use equivalency of and
Check
13
expl 13
Use equivalency of and

• Solve

Check does not exist. So -2 is not a solution.
The solution 2 does work.
14
expl 20
Write both sides as 5 raised to some power. Then
we can say the exponents are equal.

• Solve

Check
15
expl 12
Use the rules from 5.4 to simplify the left side.

• Solve

Use equivalency of and
The quadratic formula gets us x is -1 or 4. We
check both to see that -1 does not work so 4 is
our only solution.
16
expl 64

• Solve graphically.
• Using the change of base formula for the left
  side we will graph
• and see where they intersect.

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The solution to the equation is the x value where
the two sides intersect.
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Worksheet

• Using log rules to solve equations goes through
  problems step-by-step. We look at both algebraic
  and graphical solutions.
• Comments on worksheet
• 1.) Re 3 Do not rewrite as
• It will not graph the correct function.
• 2.) Re 4 Here we want the left side graphed as
• not as

5.5 homework 1 7 11 19 45 53 63 65