Deriving the Change of Base Formula for Logarithms
Date: 04/13/2007 at 22:02:49 From: Joe Subject: Formula for changing base of a logarithm- why does this work In my Pre-Calculus class we have been learning about the properties of logarithms. Since calculators have only two bases--base 10 and base e, we have learned how to change the base of a logarithm using this formula: log x b log x = ---------- a log a b I know how to use this formula, but I have no idea why it works. Can someone give me a proof explaining why this works? Thanks a lot.
Date: 04/13/2007 at 22:31:47 From: Doctor Peterson Subject: Re: Formula for changing base of a logarithm- why does this work Hi, Joe. It's not too hard to prove, though the notation can get messy! I often accidentally derive the formula in the course of solving logarithmic equations. Let's start by calling the log we're looking for y: log_a(x) = y Now we write that in exponential form: x = a^y (That is, I raised the base a to the exponent on each side of the original equation.) Now we want to solve this equation for y, using only base b logs, not base a logs. To do this, we take the log of each side: log_b(x) = log_b(a^y) Now we simplify the right side: log_b(x) = y log_b(a) To get y by itself, we just have to divide both sides by log_b(a): log_b(x) / log_b(a) = y Substituting log_a(x) back in for y we have: log_a(x) = log_b(x) / log_b(a) And we're done! As you can see, the formula is just the natural result of solving using an available log; that's why I so often get a result looking like this, and then slap myself on the head and say, "I coulda used the base-change formula!". If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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