Exponential Functions and Logarithmic Functions

Exponential Function

A function with a constant value that is changed by the term of its exponent, using the equation f(x)= ax. where a > 0 and not equal to 1, and a is the base of the function.

Here is a specific type of exponential function, dealing with the rate of growth or decay

y=a * bx--------exponent

| | |

| | | growth rate

| |

| | initial value

|

| f(x)

example:

Here is a table which represents how many folds can be created with a certain thickness of paper. As you can see in this case, the number of folds represents the possible value of the exponent, this exponent acts upon the growth rate, which is represented by the constant two.

folding paper

y=.004*(2)x

# of folds	thickness (inches)
0	.004
1	.008
2	.016
3	.032
4	.064
5	.128
6	.256
7	.512

Inverse Relations

f(x) and f-1(x) are inverses

f ( f-1(x))=x and f-1(f(x))=x

if (a, b) is on f(x) then (b,a) is on the inverse.

inverse relations are symmetric over the line Y=X

here are examples 21 and 31 from Exercise Set 4.1 on page 294.

21. Find (f x g) (X) such that h(x) = (f x g) (X)

Comparing Exponential and Logarythmic Functions:

Here is a comparison from the book, (pg 312). Here, the properties of both functions are mapped out carefully.

The inverse relationships are expressed by the functions 

y=ax

x=ay

In the first photo, the curve of the blue graph line is clearly exponential because as the x value increases positively, the y value grows exponentially.

Note: an Exponential function does not cross the x axis. Though it will come come infinitely close.

because of the inverse relationship. The properties of the logarithm rely on the coordinates (b,a).

Note: a Logarithmic function does not cross the y axis.

The second photo shows examples of comparisons between exponential and logarithmic functions when the graph follows the line y=x.

the graph on the left shows the two functions where a > 1 and the graph on the right defines a as 0 < a < 1

Here is another comparison that gives more complicated graphs and a more in depth look at reading logs.

http://www.themathpage.com/aprecalc/logarithmic-exponential-functions.htm

Here is a conversion of a log to an exponent

            |----------- output value            
a) 

log232 = 5 --------- the logarithm is the exponent

         | ---------the base remains the same    

the result: 25 = 32

here is a good table to remember for the properties of solving more complex logs.

Summary of the Properties of Logarithms

The Product Rule	loga MN=loga M + loga N
The Power Rule	loga Mp=p loga M
The Quotient Rule	loga M/N=loga M - loga N
The Change of Base Formula	logb M=loga M/loga b