



 
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 "repeatedmultiplication" concept of power to allow us to asy (for instance) that 9^{0.5} = 3, or that 10^{0.30103....}=2. Using this continuous version of the power function we can find log_{b} (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, log_{x}(25) = 2 has the unique solution x=5. Good Hunting!  


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 