Wednesday, May 18, 2011

Logarithms to Bases other than 10!

Today's class was all about logs.

Unfortunately, Instead of learning about tress and the beauty of nature we spend our time discussing Logarithms to Bases other than ten; which look a little bit more like this...

We started our class by going over homework from the previous night, 9-5, which discussed the basic ideas of logs. Just a quick review:
A common logarithm allows you to solve for x when 10^x=a.The formal definition of Logarithm of x to the base of 10 states that Y is the logarithm of x to the base o f10, or the lo off x to the base of 10, or the log base 10 of x, written y=log3 of x, if and only if 10^y=x
For example:

2=log10 of 100, because 10^2 =100.
The idea of logarithms was carried over to today's lesson, however, it answers the question "what happens if the base is not 10?" Since I know that all of you are just dying to know the answer to that, i shall now proceed and explain everything step by step.

The formal definition of Logarithm of a to the base b states that if b>0 and b is unequal to 1, then x is the logarithm of a to the base b, written x=log b of a, if and only if b^x=a
For example, because 3^5=243, you can write 5=log3 of 243. This
is read "5 is the logarithm of 243 to the base 3" or "5 is the log base 3 of 243".

These are examples of powers which have been converted from the exponential, to the logarithmic form.
3^4 = 81
log3 of 81=4
log3 of 27=3
3^2 =9
log3 of 9=2
log3 of 1=0
3^-2= 1/9
log3 of 1/9 =-2

In order to change a logarithm from the exponent to log form first the equation must be rewritten in the b^n=m form.
For example,
P= (91.028)^X
Next, divide both sides by 9
1.028^x= P/9
The base is 1.028

log1.028 of (P/9)=X

Here are examples of how to solve a logarithm to base other than 10:
Let log 7 of 49= X
  1. 7^X=49 Definition of logarithm
  2. 7^X=7^2 Rewrite 40 as a power of 7
  3. X= ? Equate the exponents
So, log 7 of 49=2

More examples from today's notes:

log10 of 100=2

log2 of 128= 7

log8 of (1/8)= -1

log9 of 3= (1/2)

log7 of 7^4/3= 4/3


Both the exponential equation 3^y=X and the log equation y=log3 of X describe the inverse of the function with equation y=3^x. In general the logarithm function with base b,g(x0=log b of x, is the inverse of the exponential function with a base b,f(x)=b^X
It is important to remember the restrictions of the graphs.
Domain: all reals
For the function y=10^x
Range: y>0
X int: none
Y int: 1
Asymptote: x axis

For the function y=10^x
Domain: x>0
Range: all reals
X int: 1
Y int: none
Asymptote: y axis

Well this about sums up the lesson that we had today! I hope this helps, even though I know that no one ever looks on this website! For more accurate and beneficial help go see Mr. Cope and take up his free time! (I'm sure he will be very happy about that! :)

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