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Solving exponential and logarithmic equations

- Section 5.5

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.

- 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

All of these functions are 1-1.

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

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

- 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

expl

- Solve
- Two different ways
- Method 1

Use equivalency of and

and simplify.

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.

expl

Use equivalency of and

- Solve
- Two different ways
- Method 1

Use the Change of Base formula on left

and simplify.

Check

If , then

- Method 2

take log of both sides

and simplify.

expl 9

- Solve

Use equivalency of and

Check

expl 13

Use equivalency of and

- Solve

Check does not exist. So -2 is not a solution.

The solution 2 does work.

expl 20

Write both sides as 5 raised to some power. Then

we can say the exponents are equal.

- Solve

Check

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.

expl 64

- Solve graphically.
- Using the change of base formula for the left

side, we will graph - and see where they intersect.

The solution to the equation is the x value where

the two sides intersect.

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

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