Question

Easy quick question: Is 1^0 1 or undefined?

Answer 1

1

Answer 2

Ok thanks, then 1 more quick question, is\[\log_{1}1 \]1 or undefined?

Answer 3

Write it as y = log...

and use the definition :-)

Answer 4

??
I'm not trying to make it into a function

Answer 5

$$\huge 1^y = 1$$

for what values of y is that^ true?

Answer 6

Oh, all real numbers

Answer 7

you can not have base of 1.

Answer 8

my guess is that ANYTHING to the power 0 is 1. A sort of "catch all cop out clause" in maths ... ? That Rubik's cube looks "interesting" ...

Answer 9

$$\huge y = log_b x $$

$$\huge b^y = x$$

Answer 10

I think you.. or someone.. had asked this question the other day..
and I was curious and found a neat answer on stackexchange.

If you try to apply change of base,\[\large\rm \log_11=\frac{\ln1}{\ln1}\]Which gives us division by zero,
so it's just a way to show that this is a problem.

Answer 11

log_1(1) = 1 but this is just a special case. In general, the base can not be 1 because y = b^x is an exponential function. If b = 1, it's no longer an exponential function

Answer 12

the base is only defined to be between 0 and 1 exclusively, or strictly bigger than 1.

Answer 13

@zepdrix ah, that also makes sense.

Answer 14

And also, for logs, the bases can be negative and 0 correct?

Answer 15

Good question! @steve816 :D

Answer 16

Because when I try to graph \[\large \log_{-2} x\] on desmos, I don't see anything.

Answer 17

What definition are you using for logs?

Answer 18

What do you mean definition?

Answer 19

What defines the logarithm of a number?

Answer 20

Just the standard definition\[\large \log_{a} x =y~~~~then~~~~~a^y=x\]

Answer 21

So starting with $$a^y = x$$ are there any restrictions on a, x, or y?

Answer 22

a: can be all real numbers
y: can be all real numbers
x: x can't be 0

Answer 23

If a and x are positive numbers ( a is not equal to 1),

then $$log_a x = y$$

if and only if $$a^y = x$$

Answer 24

So... can the base a be a negative number?

Answer 25

Not according to the standard definition.

Answer 26

Usually, when "0" or "empty set" are special cases of some calculation, you want to define it as the "default value" that make the other properties work fine (if possible).

That's why

* \(x^0 = 1\) because the property \(x^{a+0} = x^a x^0 = x^a\) is still satisfied.
* \(\sum_{i\in \varnothing} x_i = 0\), so that \(\sum_{i\in A\cup \varnothing} x_i = \sum_{i\in A} x_i + \sum_{i\in \varnothing} x_i = \sum_{i\in A} x_i\)
* \(\prod{i\in \varnothing} x_i = 1\), so that \(\prod_{i\in A\cup \varnothing} x_i = \prod_{i\in A} x_i \times \prod_{i\in \varnothing} x_i = \prod_{i\in A} x_i\)

The "default" values are found like that.