Question

soft question

Answer 1

if \(y=\log x,\ \{x,y\}\in \mathbb{R}\) can \(x\) be \(0\) or \(x<0\)

Answer 2

no!
e^y=0???? can you find such number that goes with that

Answer 3

or any base actually b^a=0!!

Answer 4

x can be only >0

Answer 5

and what about \(x<0\)

Answer 6

the same reason goes for x<0
e^y is always positive thus log x for x>0

Answer 7

the logarithm function is defined for positive numbers only, so x can not be < 0

Answer 8

this restrictions are there for the bond to exp

Answer 9

ok thanks

Answer 10

:)

Answer 11

by the way is there a logarithm for negative numbers

Answer 12

there is only one possibility that a power result in zero
that is the case 0^0 but that is a calculus problem hehe

Answer 13

it is not quite zero but not 1 either!

Answer 14

no, there is not a logarithm of negative number

Answer 15

you mean log base negative ?

Answer 16

and what about complex numbers

Answer 17

i mean where x can be taken negative

Answer 18

no cannot x>0 always

Answer 19

yes! I think that the logarithm of a complex number is defined

Answer 20

in complex theory, yes there are some stuff of that sort:)
i didn't take complex analysis yet but i do believe they do some kind of tricks around that

Answer 21

but logs take multiple values in some way! if we allow it to be complex function

Answer 22

if we define a complex number like this:
\[\Large z = \rho {e^{i\theta }}\]
then the logarithmic function is:
\[\Large f\left( z \right) = \ln z = \ln \rho + i\theta \]

Answer 23

where the subsequent additional contition holds:
\[\Large 0 \leqslant \theta < 2\pi \]

Answer 24

it is a whole other interesting place :)
complex numbers tend to solve such problem with some good tricks

Answer 25

that's right! @xapproachesinfinity