Why must the base of a logarithm be a positive

Aditya Erickson

Aditya Erickson

Answered question

2022-05-19

Why must the base of a logarithm be a positive real number not equal to 1?
and why must x be positive?
Thanks.

Answer & Explanation

morssiden5g

morssiden5g

Beginner2022-05-20Added 9 answers

By definition, log b x is the number for which, if you take b to that power, you get x. Symbolically:
b log b x = x
For example, what power do we need to raise 2 to in order to get 4? Well, it's log 2 4 = 2. What power do we need to raise 81 to in order to get 9? Well, it's log 81 9 = 0.5
Ask yourself what log 1 x means. It's the power, say p, for which 1 p = x
Unless x = 1, there is no solution, and when x = 1 any power will do, so log 1 1 is any number.
For the same reason log 0 doesn't make sense because we can't solve 0 y = x unless x = 0, and when x = 0, any power will do, so log 0 0 could be any number.
Why can logarithms only be applied to positive arguments? Well, log 2 ( 1 ) would be the power, say p, for which 2 p = 1. Hopefully, you can see that 2 p > 0 for all real numbers p

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