Log of worms

[LB]

http://www.wolframalpha.com/input/?i=log+base+1+of+-1
http://www.wolframalpha.com/input/?i=y+%3D+log+base+x+of+-1
http://www.wolframalpha.com/input/?i=y+%3D+log+base+1+of+x
I believe I have opened a can of worms.

1. What is complex infinity? I notice it appears as a vertical line on the graph of log_x(-1) where x is 1.
2. Why does it consider log₁(x) as complex infinity in all cases? Wouldn't log₁(1) be 1 because 1¹=1? My TI-84 Plus gives me a divide by zero error for log₁(1), which is even more confusing.
3. What is the simplified form of log_x(-1)? I mean, what is the other way to write it in terms of x and i?

[tition]

Note: mathematicians never use the notation log_x in the way you do.
I am assuming your notation right now, but will deny having done so unless tortured.

I guess (log_x)(y) is defined as (log_e y)/(log_e x) .
Therefore log_1 (1) = log_e (1)/(log_e 1) , which makes sense to be interpreted as 1.

3. What is the simplified form of logx(-1)? I mean, what is the other way to write it in terms of x and i?

It is not defined in a very strong sense of the word. In fact, I cannot clarify this confusion
with the notation log_x y, which is the reason why mathematicians don't use it.

[LB]

I am using the notation used by WolframAlpha, and that I have been taught. Now I am curious;

4. What notation do mathematicians use for logarithms, and why don't they use the one I have been taught?

[tition]

log z denotes ([Edit:] the principal branch of) the complex logarithm of z with base e. The principal branch log z is more correctly denoted by log_0 x.

ln x denotes real logarithm of x with base e.

[Edit:] Edit: logarithms in base other than e are not used. Mathematical texts will never use the expression "logarithm base ..."; they simply say logarithm. The expression (ln y)/(ln x) is completely unambiguous.

Define z:=x+i y

For (x,y) not equal to (number<=0, 0), define

arctg(y/x):=the angle \alpha in the range (-\pi, \pi) such that tg (\alpha)=y/x.

The above can be made into a well-defined function for x=0, y=non-zero.

Now define

log_0 z:= 1/2 ln (x^2+y^2) + i arctg(y/x)

Define

log_n z:= 1/2 ln (x^2+y^2) + i ( arctg(y/x)+ 2n \pi).

For log_n z you can think of chosing arctg (y/x) to take values in the range (-\pi +2n\pi, \pi +2n\pi).

Defining that arctg (y/x) takes values in the range (-\pi, \pi ) corresponds to cutting a ray from (0,0) to -infinity along the x axis.

You can define arctg (y/x) to take value in the range (a, a+2\pi) for any real number a, by cutting the (x,y)-plane along a ray from (0,0) to infinity along a ray at the angle \pi+a\pi . That would be denoted in mathematical language by

log_a (z)

Finally, you don't need to use a straight line in order to cut the plane in order to define arctg. For different points on the same ray, you might choose values differing by 2pi. In fact, any cut from (0,0) to infinity that does not self-intersect will work.

Last note:

in the complex numbers, under the usual convention, there is only one infinity point i.e. -infinity coincides with +infinity.

[LB]

Well, nevermind 4 then. I guess being in Pre-Calculus this is still way over my head.

[Kyon]

2. Why does it consider log1(x) as complex infinity in all cases? Wouldn't log1(1) be 1 because 11=1? My TI-84 Plus gives me a divide by zero error for log1(1), which is even more confusing.

Log1(x) is converted to: Log(x)/Log(1)
Log(1) equals zero. (Since x^0 = 1)
Log(x)/0
Depending on your machine, it may produce a division error or an infinite value.

General note
Remember that in WolframAlpha (and most notations) Log(x) equals Ln(x) (meaning it's the logarithm with base e). The following relation maybe useful when dealing with these things:
e^i*π = -1

(Keep in mind that i² = -1)

EDIT:
This post is getting messy, but what ever.

1. What is complex infinity? I notice it appears as a vertical line on the graph of logx(-1) where x is 1.

Log_x(-1) converts to Log(-1)/Log(x), when x is one, this converts to: i*π/0, which would either give some form of infinity or a division by zero error:
http://en.wikipedia.org/wiki/Division_by_zero#Formal_operations