3-2: Exponential and Logarithmic Functions (AKA Bloggerithms)

While in the land of Exponential Functions, you probably realized the stimulating graphs you created were, fascinatingly enough, one-to-one. Astounded by the implications of such a thing you probably also began to marvel at the possibilities of there existing an inverse - some might call it an evil twin - to your slope-ed friend.


Lucky for us, although intimidatingly rapid at first, our villain slows quickly. Rather than increasing its increase-tion with each increase in x and decreasing its decrease-tion with each decrease in x, this fiend would decrease its increase-tion with each increase and increase its decrease-tion with each decrease. (For a less word-ed analysis of this, see "Possibly Dangerous" on the right)

In sneaking a look at the next chapter in your book, you also probably discovered that this anti-exponent-graph is called the Logarithmic Function.

Logarithmic Function:

Although most commonly written in Logarithmic Form (top), the Logarithmic Function may also disguise itself as its counterpart in Exponential Form (bottom).
Both equations are asking what power you must raise a to in order to equal x. Stay informed. Do not become prey to its deception.

Log Graph:
Because the Logarithmic Function is the inverse of the Exponential Function, its domain, range, and asymptotes are switched. Shifting the graph up/down will change the intercept and right/left will change the asymptote and the domain.
   Domain: (0,∞)

Nautilus Shell- the perfect logarithmic spiral

   Range: (-∞,∞)
   Vertical Asymptote: x = 0
   Intercept: (1,0)

Log Properties:
    
    
Inverse Properties:
    
    
One -to- One Property:
    

The Natural Log: e
Just as it exists with exponents, the irrational number e also comes into play with logarithms.
It follows the same rules as any other logarithm but it is often written .

Log Doodles:  http://www.youtube.com/watch?v=ahXIMUkSXX0

Thats just about all there is to say about Logarithms until next section..

'Log'ing off.... Olivia Miller

3.1 Exponential Functions and their Graphs

Why hello again blog post. It hasn't been long enough.

Friends, faculty, family members and random internet travelers: today I am here to tell you about exponential functions and their graphs.

Exponential functions can be written in the form

 

y = a * b^(x+c) + d

or you may see

 

f(x)= e^x

e is a constant that is an irrational constant around 2.7183, while x is a variable.

So, let's look at some graphs

 

This is actually a giraffe. But it's a common mistake. If that's actually what you were searching for, here's a video you might like:
http://www.youtube.com/watch?v=VDhNutbXpFE

Majestic, aren't they?
But really when graphing an exponential function it's important to identify:

•Asymptotes
 Start from the asymptote being y=0. Then it is important to look at two variables; c and d. In the exponential equation stated above c and d are both capable of shifting the graph up and down the graph.

•Intercepts
The x and y intercepts are found the same way as in any function. Plug in 0 for the y value to find the x-intercept and plug in 0 for the x value to find the y-intercept.

•Whether the graph is increasing or decreasing
If x doesn't have a negative coeffecient then the graph will increase. If the graph looks something like y=a*b^(-x) then it will decrease. The graph will also decrease if  0<b<1. Assuming a is postive. If a is negative then will the graph will reflect over the x-axis, reversing everything I just said.

It's also important to note how much the graph has been stretched by the value of b. As b gets larger the graph will be horizontally compressed and the opposite as it gets smaller. And the how much the graph has been shifted needs to be taken into account.

Some very basic examples:

A practical use of exponential functions is in compound interest.

Compound interest can be expressed in the formula

                 

A=P(1+r/n)^(nt)

 

A is the balance of the account, P is principal, t is the number of years, r is an annual interest rate and n is the number of times per year that the account compounds.

 

 

The book lickers must be stopped.

Operations Research is the place to be next year.

Peter's spheres are no match for mine.

Go frustums

 

Andrew