Math CentralQuandaries & Queries


nazrul, a teacher:

If log base x of 25=2 then what is the value of x?
I am confused if the result be 5 or -5.
Please explain.
Thank you.

In one sense, both are right - and that was a very perceptive observation! However, we normally assume the base of a logarithm to be a positive number.

The reason for this is negative bases only work in cases that are too special to be of interest. Remember, we can extend the "repeated-multiplication" concept of power to allow us to asy (for instance) that 90.5 = 3, or that 100.30103....=2. Using this continuous version of the power function we can find logb (x) for any b greater than 1. (Bases between 0 and 1 can also be used though we usually don't.)

However, while (-5)2 = 25, we cannot find any real number such that (-5)x = 10, or even such that (-5)x = 5. We can't even evaluate (-5) 0.5 as a real number. If we really, really, need to we can find a complex number that is the log base (-5) of 10 [in fact an infinite set of them] but for most purposes that is a bad idea.

So for almost any purpose you would ever want to think of, logx(25) = 2 has the unique solution x=5.

Good Hunting!

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