Why can't a equal 1 in y=log(base a) x.
And why is x always positive in that?
Hmmmm. a^y=x (from the problem statement).
If a were to equal 1, then x would be 1 (1 to any power is 1). So if a is not one, but a positive number, then x has to be a positive number, if one is restricted to the real number system.
Yes, but if x is always one...why does that mean "a" can't equal one? The function would still exist.
consider the log equation
y = log1 x
in exponential form this would be 1y = x
1 raised to anything stays 1 so it is merely the line x=1
as to your second question:
suppose you look at y= log-2 x
in exponential form this is (-2)^y = x
for integers of y, (-2)^y will alternate between + and - values of x
what if y is a fraction
if y=1/2, x is undefined
if y=1/3, x is a real number
if y=1/4, x is undefined
if y=1/5, x is a real number
etc.
this is a mathematical nightmare.
furthermore, the log function is the inverse of an exponential function
those exponential functions in their simple form lie totally above the y axis,
and since the inverse function lies totally to the right of the y-axis, this explains why the base is always positive
But again, how would that show, that x cannot equal one? The function would still exist and be defined...there wouldn't be anything wrong with it, if it equalled one.
True, but it would be silly and meaningless. 1=1 is not much more than the identity postulate. X would not be a variable, it would not vary with y. Y could be anything, x would be 1. Hardly a function there.
If you study the theory of complex functions you can get a deeper insight into these matters. If you just rewrite the logarithm in base a as Ln(x)/Ln(a), then you see that the function diverges at a = 1 if x is not equal to 1.
If you know the formula:
Exp(i x) = Cos(x) + i sin(x)
you can understand the definition of the logarithm for complex values. E.g. Ln(-1) = i pi, because Exp(i pi) = -1.
Note that there is an ambiguity of
2 pi i n where n is an integer in the definition of the logarithm. If you go around the origin once the value increases by 2 pi i.
To define a single valued logarithmic function unambiguously, one has to specify a so-called branch cut in te complex plane. If you cross that line the value jumps by 2 pi i, to make sure that the value of the logarithm stays the same if you go around the origin once. The branch cut is conventionally chosen to be the negative x-axis.