Exponential Equations I

Exponential Functions

The Exponential Function Base e

The Logarithm Function

Logarithmic & Exponential Equations II

Applications:Growth & Decay






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.









Example 2: Solve for x. Check # b.







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?




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.




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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