Solving Exponential Equations

Remember, when you are solving an equation, you are

trying to the find the value of the unknown that makes the left

side equal the right side

There are 3 methods used to solve an equation:

1. Inspection

2. Guess and Check

3. Operations

Solve the following:

x + 7 = 10 by inspection

x = 3

4 – 3 = x by inspection

x = 1

Solve the following:

x + 17 = 35 by guess and check

x = 18

42 = x

x = 16

Solve the following:

x2 = 9

x = 3 or -3

The last kind of example represents the lesson today.

The variable could be the exponent…

Solve the following

2x = 32 x = 5

NOTICE, we can re-write 32 as a power of 2 (if you can not inspect)

2x = 25

Since the bases are equal, the exponents must be as well.

We will use this re-writing technique when we can not solve

by inspection

4x + 1 = 2x - 1

4x + 1 = 2x - 1 Since 4 = 22

(22)x + 1 = 2x - 1 • P of P

• Dis Prop22x + 2 = 2x - 1

Since the bases are equal, so are the exponents, therefore we can create a new equation using just the exponents.2x + 2 = x - 1

-x -x2x + 2 = x - 1

1x + 2 = - 1

-2 -2

x = -3

4-3 + 1 = 2-3 – 1 ?

Does x = -3 work?4x+ 1 = 2x - 1

4-2 = 2-4

1 = 142 24

YES!

1 = 116 16

93x + 1 = 27x Since 9 = 32, and 27 = 33

(32)3x + 1 = (33)x

• P of P

• Dis Prop36x + 2 = 33x

Since the bases are equal, so are the exponents, therefore we can create a new equation using just the exponents.

6x + 2 = 3x -3x -3x

3x + 2 = 0

Exponential Decay

Iodine – 131 is used to determine if a person has a thyroid deficiency.

• The thyroid (from the Greek word for "shield", after its shape) is one of the larger endocrine glands in the body. It is a double-lobed structure located in the neck and produces hormones, principally thyroxine (T4) and triiodothyronine (T3), that regulate the rate of metabolism and affect the growth and rate of function of many other systems in the body. The hormone calcitonin is also produced and controls calcium blood levels. Iodine is necessary for the production of both hormones. Hyperthyroidism (overactive thyroid) and hypothyroidism (underactive thyroid) are the most common problems of the thyroid gland. Specialists are called Thyroidologists.

Iodine – 131 is injected into the patient.

A healthy thyroid will absorb all the iodine.

An exponential equation will tell them how much iodine to look for.

AL = AO 12

t8.2

AL = the amount of iodine left

AO = the original amount of iodine

t = the elapsed time in days

How long should it take 80mg of iodine to decay into 5 mg?

Homework

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