Another Logarithm Problem
Date: 10/2/95 at 12:32:32 From: David Farnell Subject: log question We are having a very difficult time with this problem. Could you help us out? log 69= log 69 b 10 _______ log b 10 We are supposed to prove this for every positive number b but b cannot equal 1. Thank you for your help. David Farnell, Math/Science Teacher North River School District
Date: 10/2/95 at 13:8:32 From: Doctor Andrew Subject: Re: log question Why don't we try this as generally as we can. The general rule is (I'm using "_" to indicate subscripts and <> to indicate not equal): log_b a = (log_c a) / (log_c b) for a > 0, b,c > 0 and b,c <> 1. If you can prove this, your example is definitely true. All you really need for this proof is a little knowledge about exponents and the definition of a logarithm which is: x = log_b a if and only if (written iff) b^x = a I'll get you started. Let a >= 0 and b,c > 0 and b,c <> 1. Now in general if (thing1 iff thing2) and (thing2 iff thing3) then (thing1 iff thing2). Keep this in mind when going about the proof. What you want to show in the end (what you want to be at the ends of this chain of if and only ifs) is: x = log_b a iff x = log_c a / log_c b. I think the easiest way to do this problem is to start with the righthand side. What can you say that is true if and only if x = log_c a / log_c b? Good luck! If you want some help on the next step, please write us back. -Doctor Andrew, The Geometry Forum
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