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
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