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What is Log?


Date: 26 Feb 1995 22:46:28 -0500
From: charley
Subject: Math questions

Hi,

My name is Yutaka Charley and I'm in the 5th grade at PS150Q in NYC.

What's 4 to the half power?

What does log mean?

Thank you.

Yutaka


Date: 27 Feb 1995 21:54:12 -0500
From: Dr. Ken
Subject: Re: Math questions

Hello there!

I'll address your second question, the one about Logs; and my
colleague and buddy Ethan has promised to answer your first question, 
the one about 4 to the 1/2 power.

Here's the definition of Log: 

         b
  If   a  =  x,    then   Log (x) = b.
                             a

When you read that, you say "if a to the b power equals x, then the Log 
(or Logarithm) to the base a of x equals b."  Log is short for the word
Logarithm.  Here are a couple of examples:  Since 2^3 = 8, Log (8) = 3.
                                                              2

For the rest of this letter we will use ^ to represent exponents - 
2^3  means 2 to the third power.

To find out what Log (25) is, we'd ask ourselves "what power do you raise 5
                    5
to to get 25?"  Since 5^2 = 25, the answer to this one is 2.  So the
Logarithm to the base 5 of 25 is 2.

Whenever you talk about a Logarithm, you have to say what base you're
talking about.  For instance, the Logarithm to the base 3 of 81 is 4, but
the Logarithm to the base 9 of 81 is 2.  

Here are a couple of examples that you can try to figure out:  What is the
Logarithm to the base 2 of 16?  What is the Logarithm to the base 7 of 343?
How would you express the information, 4^3 = 64, in terms of Logarithms?
_______________

Now that you have done Logarithms I will take over for my buddy
Ken and talk about fractional exponents.

To help explain fractional exponents I need to teach you one neat
fact about exponents:

        3^4 times 3^5 equals 3^(4+5) or 3^9 

This will be very important so I will show a few more examples.

       4^7 times 4^10 equals 4^17
       5^2 times 5^6 equals 5^8

Now let's get to fractional exponents.  Let's start with 9^(1/2).

We know from our adding rule that 9^(1/2) times 9^(1/2) is 9^(1/2 + 1/2), 
which is 9^1; so whatever 9^(1/2) is, we know that it times itself has to
equal nine.  But what times itself equals 9?  Well 3, so 9^(1/2) is 3.

All fractional exponents work this way.  Lets look at 8^(1/3).  Again,
8^(1/3) times 8^(1/3) times 8^(1/3) is 8^(1/3 + 1/3 + 1/3), which is
8; so we need to know what times itself three times is 8.  That is 2.

So now look at your problem, 4^(1/2).  We know from experience
that this means what number times itself is 4?  That is 2, so 4^(1/2)
equals 2.

 Hope that helps,

- Ken "Dr." Math
and Ethan Doctor On Call
    
Associated Topics:
Elementary Number Sense/About Numbers
Middle School Logarithms
Middle School Number Sense/About Numbers

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