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Exponents

Exponential Equations I

Exponential Functions

The Exponential Function Base e

The Logarithm Function

Logarithmic & Exponential Equations

Applications:Growth & Decay

Review&Test

 

jdsmathnotes

 


 UNIT 5 : EXPONENTIAL & LOGARITHMIC FUNCTIONS

 LESSON 2: EXPONENTIAL EQUATIONS

 

Exponential Equations:

 

Definition: An exponential equation is an equation where the variable is in the exponent.

 

Examples of Exponential Equations: 2x = 64; 92x 1 = 729; 500(1.02)x 1 = 897.56

 

Theorem: If ax = ay, then x = y. In words: If an exponential equation has the bases equal, then the exponents must be equal.

This theorem gives us our strategy for solving exponential equations, namely convert each side of the equation to a common base.

 

Text Box: Steps for solving:
	Convert both sides of the equation to a common base
	Isolate the power containing the exponent
	Equate the exponents using the above theorem
	Solve the resulting equation

 

 

 

 

 

 

 

 

Note: Exponential equations may also be solved by taking logarithms of both sides of the equation. This method will be discussed in lesson 6

 

 

Example 1: Solve for x.

Solutions:

 

 

 

 

 

 

 

Example 2: Solve for x. Check # b.

 

Solutions:

 

 

APPLICATIONS:

 

Exponential Growth and Radioactive Decay are applications of exponential equations.

 

Example 1:

 

A bacteria culture doubles in size every 10 minutes. Its growth is measured by the following formula:

 

, where

        A is the number of bacteria after the given time frame

        A0 is the starting number of bacteria

        2 is the growth factor

        t is the total time elapsed in the experiment

        d is the doubling period

 

How many bacteria will there be in the culture after 1 hours if there were 20 bacteria in the original culture?

 

Solution:

 

A = ?

A0 = 20

t = 1 h = 90 min

d = 10 min.

 
 


 

 

 

 

 

 

Therefore there will be 10 240 bacteria in the culture after 1 hours.

 

Note: The half-life of a radioactive substance is the period of time a given amount will decay to half of its original amount.

 

Example 2:

The half-life of radioactive radon is 4 days. I t decays according to the formula below:

, where

        A is the mass remaining after the decay period

        A0 is the original mass of radioactive material

        is the decay factor

        t is the total time elapsed

        h is the half-life of the material

 

If the amount remaining after 40 days is 6.5 g, calculate the original amount.

Solution:

 

 

A = 6.5 g

A0 = ?

t = 40 days

h = 4 days

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Therefore the original mass was 6656 g.

 

Note: An equivalent formula for radioactive decay is:

 

 

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