What is the graph of y = log base 1 x?

[JinM]

I know that logarithms to the base 1 is undefined, due to the reason that:

\log_1{x} = \frac{\log_a{x}}{\log_a{1}}

And this leads to divison by zero, which is undefined.

There was a question in one of my textbooks that asked describe the graph that results if y = \log_1{x}. Is such a graph even possible?

If I switch this logarithm to exponential form I would get:

1^y = x

Now, is it possible that the graph could be y = 1 and x = 1? Since 1^y = x^1, y = 1 and x = 1.

Or is it the point of intersection of these two lines? If not, what is it?

Thanks.

 

[HallsofIvy]

Well, not "the" graph. You have two "functions" there and so two graphs. Yes, if y= log₁(x) then x= 1^y= 1 for all x. The reason I put "functions" in quotes above is that this is not actually a function: strictly speaking, log₁(x) is not a function. But it is a relation and its graph is the vertical straight line x= 1.

y= 1^x= 1 is a function: its graph is the horizontal straight line y= 1.

 

[snipez90]

Well not even going into division by zero when considering a change of base, you should recognize that 1 raised to a power won't get you x, a variable.

1 raised to any power, whether fractional, negative, or worse, is just 1. So the graph will look like x = 1.

 

[5inister]

Wouldn't the expression y=log₁1 (or y=log₁(x);x=1 for all x) be undifned too as log_n1 is always zero; So y=log₁1 has both zero and one as an answer?