Estimating Logarithms without Using a Calculator
Date: 07/28/2005 at 13:02:30 From: Amanda Subject: log problem I am trying to find the following answer without a calculator: log_4 12 From reading respones from Dr. Math, I believe the statement to read: what is the power(exponent) that raises a base of 4 to make 12. I can come to a broad conclusion that the answer is between 1 and 2. Other than that I have the problem 4^x = 12. How do I solve? Thanks
Date: 07/28/2005 at 14:53:48 From: Doctor Douglas Subject: Re: log problem Hi Amanda. Excellent work so far. Indeed you are solving 4^x = 12 for x, and it is clear that 1 is too small and 2 is too big. A good way to continue proceeding is to factorize 12: 12 = 4 * 3. So we can write 4^x = 4 * 3 Taking the log of each side, we get x*log(4) = log(4*3) = log(4) + log(3) and we can solve for x as x = [log(4) + log(3)]/log(4) = 1 + log(3)/log(4) which is about as simple as one can get without resorting to a calculator to compute the logarithms. Now it gets a little more complicated to proceed with finding a numerical answer if we are not permitted the use of a calculator. Here's a way to do so, where I assume that the logs are taken in base 10: log(3): Consider powers of 3. 3^2 = 9, which is close to 10^1. Hence 2*log(3) is approximately equal to 1*log(10) = 1, or log(3) :=: 1/2, where I use the colons to indicate "approximately equal to". For more accuracy, we can take higher powers of 3 until we find another one that is close to a power of 10: 3^21 :=: 10^10, so that log(3) = 10/21 = 0.47619. log(4): We do the same with powers of 4: 4^5 :=: 10^3, so log(4) :=: 3/5 = 0.6, to pretty good accuracy. I say pretty good accuracy because 4^5 = 1024 and 10^3 = 1000 only differ by 2.4%. In fact we can't do much better until going all the way to 4^98 :=: 10^59, whereupon log(4) :=: 59/98 = 0.6020408... The calculations above require nothing more than simple arithmetic and a knowledge of the rules of logarithms, although I admit that the multiplication can be painstaking. So our estimates are x = 1 + log(3)/log(4) :=: 1 + (1/2)/(3/5) = 1.83... by computing 3^2 and 4^5 :=: 1 + (10/21)/(59/98) = 1.791... by computing 3^21 and 4^98 Using a calculator, I obtain log(3) = 0.477121255.... and log(4) = 0.602059991..., and a final answer of x = 1.792481250360578... This shows that our estimates aren't too bad. Remark: we could have just started from powers of 12 in the beginning: 12^5 = 248832 :=: 262144 = 4^9, so an estimate for our answer is x = log(12)/log(4) = 9/5 = 1.8, which is not too bad. However, if we do it this way then we have to be able to recognize when a power of 12 is in fact close to a power of 4. It's a lot easier to do these computations by using base 10 as an intermediary, since it is very easy to recognize the powers of 10. To do this efficiently, it was helpful in the beginning to factor the original 12 into smaller numbers to reduce the amount of multiplicative arithmetic. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/
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