3.3 Properties of Logarithms

Solving logarithmic equations requires the use of four properties.

Addition of Logarithms (the base a must be a positive number other than 1)

When adding two logarithms of equal bases, you can combine the two logs into one log with x and y being multiplied together.

Ex.

Subtraction of Logarithms (the base a must be a positive number other than 1)

When subtracting two logarithms of equal bases, you can combine the two logs into one log with x being divided by y.

Ex.

Exponents/Coefficients of Logarithms (the base a must be a positive number other than 1) 

If there is a coefficient b (which is a real number) of a log, it can be moved to be the exponent of x and vice versa.

Ex.

Change-of-Base Formula (a, b and x are positive real numbers. a and b are not 1)

If you have a log base a of x, you can change the base of the logarithm to any number that suits your needs with the log of x over the log of a. This formula is very useful for situations in which you need to calculate the numeric value of a log, but the base isn't 10 or e, which are the only two calculable on a calculator. 

Ex.

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

2.6: Rational Functions

A rational function is a function that can be written as:  where N(x) and D(x) are both polynomials, and where D(X) is not the zero multiplicity.

Ex. 

When f(x)=0, the graph of a rational function is an x-intercept, this indicates that N(x)=0 and D(x) is a nonzero integer.

Ex. ; f(x)= 0 where , such as at x=1

When D(x)=0, the graph of a  rational function produces a vertical asymptote, this an undefined line which the graph will never cross.

Ex. ; When x=2, D(x)= 8-8=0, causing f(x) to be undefined and producing a vertical asymptote.

The graph below shows a rational function, the graph has a vertical asymptote on the line x=0. As you can see, the graph approaches the line x=0, but never crosses it.

Graphs of rational function can produce y-intercepts at f(0), sometimes there is a vertical asymptote at the line x=0, and therefore, the graph has no y-intercepts. This is the case in the graph above.

Ex. ; (0,-1/8) is a y-intercept of this rational function.

End Behavior

There are 3 cases for the end behavior of rational functions:
1) The degree of the leading term of N(x)>the degree of the leading term of D(x)
2) The degree of the leading term of N(x)=the degree of the leading term of D(x)
3) The degree of the leading term of N(x)<the degree of the leading term of D(x)

Each of these cases has a distinct end behavior.

1) When the degree of the numerator is larger, the end behavior of the rational function is: 

Ex.  when x gets very large i.e. 1 trillion, or very small i.e. -1 trillion, no matter what the degrees of the non leading terms are, the numerator will always trump the denominator. In this case, very large negative numbers will produce positive values and approach infinity, while very large positive  numbers for x will produce negative values and approach negative infinity. In this case, the limits will always be either a positive or negative infinity.

2) When the degree of the numerator is the same as the degree of the denominator, the end behavior of the rational function is a ratio of the leading coefficients of N(x) over the leading coefficients of D(x).

Ex. no matter what the value of x is, the value can cancel, leaving only the coefficients, which simplify to a ratio of -1/3. The limit notation for this function would be:

3) When the degree of the denominator is larger than the degree of the numerator, the limit notation is always:

This is because when you plug in a very large positive or negative number into any rational function that matches this criteria, such as , no matter what, as the magnitude of the x value increases, the value of the function will always approach zero. ***This case could also approach a different number, in rare cases, if the function had an addition or subtraction outside of the fraction which shifted the entire graph up or down.***

In cases 2 and 3, horizontals asymptotes are created, because the limits of x approach a number, such as 0, or -1/3, but never reach them. The graph above shows a horizontal asymptote at the line y=0.

A couple extra things to remember:
- The graphs of rational functions are mostly discontinuous, this means that the graph in disconnected at atleast one point. These points are usually lines, which form asymptotes. This is an important fact to remember because it may show up on a test. ;)
- If the limit of x is not positive and negative infinity, you must take this into account, often you may have indicate limit coming from the right or left side, approaching a number.
Ex. is x approaching 3 from the right, and is x approaching 3 from the left.
-If any x value every makes N(x)=0=D(x), there will be a hole is the graph, this value is called in-determinant and is something that must always be shown when drawing any graph of a rational function.

-C. Baller
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||///////////////////////////////////////////////////////

http://www.youtube.com/watch?v=pjrI91J6jOw&list=UUOGeU-1Fig3rrDjhm9Zs_wg&index=3

//////////////////////////////////////////|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Section 2.4: Complex Numbers

Imaginary unit i is used to represent the square root of a negative number, where i equals the square root of -1. 

Complex numbers are in standard form a + bi, where a and b are real numbers.

To be considered a complex number, the expression must be in the exact form a + bi. Examples of complex numbers in standard form:

When raising i to various powers, follow this simple pattern:

Complex numbers cannot be in the denominator, so in order to get it into the top, multiply the whole fraction be the conjugate (a - bi).

That seems to be about it for imaginary and complex numbers! 

- Jessica

Synthetic Division and Long Division

Synthetic division and long division can be used for dividing, factoring, and finding the zeros of polynomials. 

Long division of polynomials is basically the same as long division of numbers. You start with a polynomial, say x^4-2x^3+x-4. Lets say you were dividing it by x+2. You set the two polynomials up as you would a basic long division problem, with the dividend under the division sign and the divisor to the left. Make sure to put in a placeholder if a certain degree is missing. In this case, you would add 0x^2.

LaTex leaves a blank space, but just imagine a 0x^2 there. The first step is to look at the first term in the dividend, in this case, x^4. You then look at the first term in the divisor, x. Determine how many times it goes into the first dividend term and write it above the term, matching exponents. For instance, you determine that x goes into x^4 x^3 times, so you write x^3 above 2x^3. Then you have to multiply any other terms in the divisor by x^3 as well, so you have 2x^3. Subtract the entire term (x^4+2x^3) from the first two terms, being sure to distribute the negative.

You then repeat the entire process again with the remainder until you can no longer multiply the divisor by a whole number to factor into the remainder. In this example, you end up with x^3-4x^2+8x-15, with a remainder of 26.

Synthetic division does the same thing, just by a different method. You begin by writing out all of the coefficients of the polynomial in order, leaving a zero as a placeholder for any missing exponents. To the left, write the zero of the term you are dividing by. For instance, we're dividing by x+2, so if x+2=0, then the zero would be -2, so we have -2 to the left.

Start by dropping the first term straight down to below the line. Then multiply that term by the number to the left, and write that below the second number. Add the two together and write the sum below the line. Then continue to multiply that number by the number to the left, writing it under the next number, and adding until you run out of numbers. The resulting numbers are the coefficients of the quotient, each with x to 1 less degree than the coefficient above it. For instance, we have x^3-4x^2+8x-15, with remainder 26. This is the exact same problem as before, just solved with a different method.

Synthetic division can also be used to factor polynomials. Say you have the polynomial x^3-x^2-4x+4. You can set up an equation, dividing that by one of its zeros. You try 1.

You know that factoring a polynomial has worked if you end up with a zero at the end, meaning there is no remainder. You can now rewrite the expression as (x-1)(x^2-4). You can even take it one step further and factor your result. This time you divide it by 2.

It worked again! So now you have a completely factored polynomial that can be written as (x+2)(x-1)(x-2). But how do you know which numbers to try dividing by? There are infinitely many possibilities, after all. One way to narrow down the options is to divide all of the factors of the last term by all the factors of the first term. In that last polynomial, that would leave you with 4,-4,1,-1,2, and -2. They aren't necessarily all zeros, but it considerably narrows down the options.