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Solving exponential and logarithmic equations. Sections 6.2, 6.3, 6.5. The details about functions. If a relationship is a function , then for every x value, there is only one y value. Another way to put this is, if for some numbers u and v , then . . Solve 5 = x – 2.

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the details about functions
The details about functions
  • If a relationship is a function, then for every x value, there is only one y value. Another way to put this is, if for some numbers u and v, then .

Solve 5 = x – 2.

We do it by applying the function

f(x) = x + 2. Because of the above statement, we can apply the function to both sides (getting x = 7) and the equality holds. We’ll do the same with logarithmic and exponential functions.

the details about one to one
The details about one-to-one
  • 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 .

Solve . Since

is one-to-one, then we can say x = 4.

Solve . Since

is not one-to-one, then we cannot say x = 4. What else could x be?

rules

These rules will help us solve equations.

Rules
  • 1. 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.)
  • 2. 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

rules6

These rules will help us solve equations.

Rules
  • 3. 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.)
  • 4. Since is a function, we can say that if , then . (Here M, N, and a are positive real numbers, and a is not 1.)
slide7
Remember and undo each other. They are inverses and so whatever the one does, the other will undo. The four rules are connected by this inverse relationship.
  • We’re solving equations in these sections. 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 6.4
slide8
expl:
  • Solve
  • Two different ways:
  • Method 1:

Use equivalency of

and

…and simplify.

slide9

Use the inverse relationship between

and

.

  • Method 2:

Rule 3

Apply

to both sides.

The logarithm function undid the exponential function and left 4w on the left.

slide10

Use equivalency of

and

expl:
  • Solve
  • Three different ways:
  • Method 1:

Use the Change of Base formula on left

…and simplify.

Check

slide11

Rule 4

“take log of both sides”

  • Method 2:

…and simplify.

slide12

Rule 4

“take natural log of both sides”

  • Method 3:

…and simplify. Unbury the t.

expl 7 pg 471
expl: #7, pg 471
  • Solve
  • Method 1:

Use equivalency of

and

Check

slide14

The plan is to combine the logs so deal with the 3 on the first term.

  • Method 2:

Use equivalency of

and

… and unbury the x.

expl 10 pg 471
expl: #10, pg 471

Use the rules from 6.4 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.

expl 89 pg 455
expl: #89, pg 455

Use equivalency of

and

  • Solve

The base b must be positive and not 1.

Check:

does not exist.

So -2 is not a solution. The solution 2 does work.

expl 48 pg 471
expl: #48, pg 471
  • Solve graphically.
  • Using the change of base formula for the left side, we will graph
  • and see where they intersect.
worksheet
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
worksheet21
Worksheet
  • “Logarithmic and exponential applications: Intensity of sound” introduces the loudness of sounds, measured in decibels, as a neat little example of the logarithmic function. We practice solving equations using this application.
  • 6.2 homework: 43, 50, 52
  • 6.3 homework: 85, 87, 91, 101, 111, 115, 122
  • 6.5 homework: 1, 5, 9, 15, 17, 29, 37, 47, 49