5.4 Properties Of Logarithms

There are three very important laws of logarithms

Laws of Logorithms

Log a (uw) =log a (u) + log a (w)

Log a (u/w)= log a (u) - log a (w)

Log a (u^c) = c log a u. For every real number c

But remember, logarithms can be sneaky, and aren't always what you think. For example

Log a (u+w) Does NOT = log a u + log a w

Also

Log a (u-w) does NOT = log a u- log a w

How to use the laws of Logorithms

There are two ways to express Logorithms. One is as one Logorithm and the other is in terms of variables.

In order to change from one Logorithm into variables you would take the exponent of the variable and make it the coefficient in front of log.

For example you would change

log a x^3 into 3 log a x.

Remember, the whole point is to use the three laws of Logorithms.

The first step in changing more complex problems is to make sure that you have taken out all the multiplication and division, and replaced it with addition and subtraction.

The next step is to change the exponents to the coefficient, as we did earlier.

Now, for expression as one Logorithm.

The first step is just the inverse of last time. Here we change

3 log a x into log a x^3.

Next we will do the opposite again. Changing. From addition and subtraction into multiplication and division.

Next we solve logarithmic equations.

Log a (2x+3)= log a 11+ log a 3

Now remember- Log a (uw) =log a (u) + log a (w)

I told you Logorithms are sneaky

Because of law one the next step is

2x+3=33

So

X=15

How to solve ane equation like

Log 2 (x+2)= 4

Since you can't just get rid of the log 2's you have to do something with them.

It is important to remember that log 2 of 4 = 2^4. All you have to do is change it to exponential form.

So

X+2= 2^4

X+2=16

X=14

Graphs of logarithmic equations

When graphing log a x^2 it is the same except is reflected about the y axis, in addition to being where it is now.

Matthew "Seaglass" Silbergleit

Hey so after some great strategy by Hannah and I, I have yet again gotten the blog post.  Today in class we learned about Logarithms.Oh and thank you Mr.Wilhelm for finally changing the background... 

In an equation...

we learned that x is being expientiated and to solve this you need to log it. 

In this chapter we are learning about exponential functions, and long and behold they are one-to-one.  And what do we know about one-to-one functions??? They all have inverses!!! The inverse to exponential functions are logarithmic functions.

We know this because inverse functions are reflected over X=Y. 














Now the KEY TO EVERYTHING (also called the Definition of loga in the book) is that...

The first equation is in exponential form when the second one is in logarithmic form.

Changing equations from exponential form to logarithmic form and vice versa is fairly easy.  All you have to do is this...

(Index means exponent)










Now to solve logarithmic equations is pretty easy too.  The easiest way to do it is to change it to exponential form and we already know how to solve those.


    Given 

             Change to exponential form

               Simplify

               add 4 to both sides 






Now there are two different types of logs. 


The common log 


The natural log (the one we are gonna use most)





In the homework there were equations that looked like this
which are far less confusing than you would think. 


All this is is another way to write exponential functions.
This just means that.. 



or





Finally we learned about the graph of logarithmic functions.

Changes-
a-vertical stretch
b-higher number makes the graph grow slower
c-Shifts left/right
d- shifts up/down


So that's pretty much it for logarithms.  Thanks again to Hannah for being a very sneaky and loyal friend.
And everyone remember that finials are approaching and there is a good chance that we shall all fail.

-Jennifer Kendall

REAL 5.1 and 5.2 Exponential Functions and The Natural Exponential Function

Back again!

So with exponential functions, unlike with polynomial and linear functions, the constant term is the base and the exponent is the variable.

f(x)=a^x for every x in , where a>0 and a1

as opposed to

This is the graph of :

The way to write an exponential function is

x is the exponential variable

b is the constant base

d is the horizontal asymptote (it will move the function up and down)

c will move the function left and right

a stretch or compress the graph, and is also the y-intercept.

when x=0 and c=0, =1, therefore the y-coordinate at x=0 wil be a (plus d of course)

if a=1, then the y intercept will always be 1 if d=0 and c=0 as well

when an exponential graph is to a negative power (or fraction, they're the same thing) then the graph is flipped horizontally

is the same as .

Here are some examples of solving exponential equations.

Another thing we learned about......

bumbumbumbumbumbumbumbumbumbumbumbum

dee dee dee dee daaaaa da daaaaa da daaaaaaaaaaaaa!!!!!!!!!!

Compound Interest Formula!!!!!!

A=present value

P=principal (initial value)

r=interest rate

n=#of compounding periods per year

t=#of years

I think all of this should be apparent. ;)

the only other thing that is really in 5.2 is not "e", but . looks much better.

is a magical number that somehow shows up in trees.

is the "natural number" and is very important. For those of you in APUSH, is the Ben Franklin of logarithms.

I hope you all had a great time, made fond memories, the natural exponential function is , and Good Night!!!

5.1 Exponential Functions

I was just too tired to write this blog post. I am sorry that I could not supplement you with more maths knowledge. Maths is forever.

NNNNNNNNNNNNOOOOOOOOOOOOOAAAAAAAAAAAAAHHHHHH!!!!!!!!!!!

3

3

4.5: Rational Functions

Hello friends! Let's learn about rational functions.

Let's start with a definition:

The domain of a rational function consists of all real numbers EXCEPT the zeros of the

denominator. Because we all know what happens when there's a zero

in the denominator...

Anyway, the cool thing about rational functions is they make these graphs call

ed hyperbolas. A hyperbola is a graph that comes REALLY REALLY REALLY close to a line, but never hits it. This line is called an asymptote. This kind of makes me sad. Like, poor function...all it wants to do is intersect the asymptote, but it never will. If this confuses you, here's some ways to think about it:

<<< Mr. Wilhelm walking towards a wall

Make sense? Good. Okay so now we move on to types of asymptotes: vertical and horizontal.

Vertical asymptote: The line x = a is a vertical asymptote for the graph of a function ƒ if

ƒ(x) ➞ ∞ or ƒ(x) ➞ -∞ as x approaches a from either the left or the right.

Horizontal asymptote: The line y = c is a horizontal asymptote for the graph of the function ƒ if

ƒ(x) ➞ c as ƒ(x) ➞ ∞ or ƒ(x) ➞ -∞.

__________________________

Theorem on Horizontal Asymptotes:

Let...

where

.

1. If n

2. If n=k, then the line y=a_n/b_k, or the ratio of the leading coefficients is the horizontal asymptote for the graph of ƒ.

3. If n>k, the graph of ƒ ha no horizontal asymptote. Instead, either ƒ(x) ➞ ∞ or ƒ(x) ➞ -∞ as ƒ(x) ➞ ∞ or as ƒ(x) ➞ -∞.

__________________________

GRAPHING RATIONAL FUNCTIONS!

Guidelines to follow:

1. find the x-intercepts - find the real zeros of the numerator and plot them on the x-axis

2. find the vertical asymptotes - find the real zeros of the denominator, and for each real zero a, sketch the vertical asymptote x = a

3. find the y-intercept, if it exists - set the function to zero - ƒ(0) - and plot the point (0, ƒ(0)) on the y-axis

4. find the horizontal asymptote - apply the theorem on horizontal asymptotes, then sketch the horizontal asymptote

5. determine if the horizontal asymptote intersects the graph. the x-coordinates of the points of intersection are the solutions of the equation ƒ(x)=c, where y=c is the horizontal asymptote. if they exist, plot them

6. based on all of the clues you found above, finish the graph. if necessary, plug in a number for x to find where the graph is places in relation to the asymptotes. keep in mind that if one curve goes down, the one across from it will go up, and vice versa, unless there is an even multiplicity, in which case they will both go the same direction.

HOLES

Sometimes, it is impossible for a graph to include a specific point, even though it runs straight through it. This happens when parts of the numerator and denominator cancel out. For example,

In this case, even though there is a hole at x=1. To graph this, place an open dot on the coordinate where the hole occurs in the graph to show that it is not a solution.

OBLIQUE ASYMPTOTES

When the numerator has a degree ONE greater than the degree of the denominator, it has an oblique asymptote. This sounds a lot more complicated than it is. Because the function does not have a horizontal asymptote, it instead has an oblique one, which is usually like a diagonal line as an asymptote. To find the slope of this line, all you have to do is use long division to divide the numerator by the denominator.

Okay, that's pretty much all you need to know about section five!

I HOPE YOU ALL KICK ASYMPTOTE ON MONDAY'S TEST

- Olivia Darany