log base 1 (1 Viewer)

[integral95]

Yeah sorry if this was asked before, but this just crossed my mind.



So what I'm getting is that this particular log is only defined at x = 1?

But if we use the change base formula



We can see that it's not defined.

So does this even exist?

 

[InteGrand]

I think it wouldn't make sense or any use to define log to base 1. Its graph would essentially be the vertical line x = 1. Staying in the world of real numbers, 1^{t} = 1 for all real t (so raising 1 to any positive power can only yield 1 as an output).

For x > 0, log₁ x (which is the number 1 needs to be raised to to get to x) could only ever be defined for x = 1 (because only if x = 1 can we ever raise 1 to any power to get to x), and if x is 1, we'd have infinitely many values for this logarithm (any real number, since 1 raised to any real number will yield 1 as output). This is why the graph would be the vertical line x = 1, and wouldn't be much use I think. (It is generally left undefined I think.)