Explicit InversesDate: 04/21/99 at 02:00:55 From: Sidney Yin Subject: Solving a variable to a power of itself Is there a way of solving a question like this: x^x = 5 without graphing or guessing? Thanks. Date: 04/21/99 at 09:57:51 From: Doctor Mitteldorf Subject: Re: Solving a variable to a power of itself Dear Sidney, In school, you learn to expect that every function has an explicit inverse. But this is not usually true. Many more functions don't have explicit inverses than do. Write down something as simple as x+sin(x) = y and you can't solve for x. Fifth degree polynomials and higher don't have explicit inverses. a^x + b^x = y doesn't have an explicit inverse. The fact that the inverse isn't explicit doesn't really mean very much. It just means that the function isn't common enough to have its own name. The procedure for solving it isn't necessarily any more difficult than the procedure for calculating named functions; for example do you have any idea what your calculator has to go through to calculate ArcTan(x)to 10 digits? Here's one way to solve x^x = 5: search for a procedure that you can repeat over and over that will get you closer to the answer each time. It gives you confidence to start with a procedure that you know will stay on the answer once you've found it. So start with the equation you have and see if you can work with it: x^x = 5 take log of both sides x log(x) = log(5) x = log(5)/log(x) Now start with a guess, say 2.1 since we know that 2^2 = 4. The next guess is x = log(5)/log(2.1) = 2.169 Do it again: x = log(5)/log(2.169) = 2.078 Repeating the process gives 2.199, 2.041, 2.255 ... I was hoping that this procedure would home in on the answer, taking us closer and closer each time. But it doesn't - it gets a little farther away each time. But notice that each time it jumps across the solution: the answers alternate between being too big and being too small. This suggests an idea: average them in pairs. In other words, if the first guess is 2.1, the second guess won't be log (5)/log(2.1) but the average of this number with 2.1. This gives first guess = 2.1 second guess = 1/2 (log(5)/log(2.1) + 2.1) = 2.1346 third guess = 1/2 (log(5)/log(2.1346) + 2.1346) = 2.1285 fourth guess = 1/2 (log(5)/log(2.1285) + 2.1285) = 2.1295 This is already very close to the solution, 2.12937248276... You can write a computer program to repeat this process in just a few minutes. Even with a programmable calculator, the program you have to write is quick and easy - running it 5 or 6 times ("iterations") brings you as close as you'll ever need to be to the solution. - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ |
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