## mathmath333 one year ago soft question

1. mathmath333

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

2. xapproachesinfinity

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

3. xapproachesinfinity

or any base actually b^a=0!!

4. Michele_Laino

x can be only >0

5. mathmath333

and what about $$x<0$$

6. xapproachesinfinity

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

7. Michele_Laino

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

8. xapproachesinfinity

this restrictions are there for the bond to exp

9. mathmath333

ok thanks

10. Michele_Laino

:)

11. mathmath333

by the way is there a logarithm for negative numbers

12. xapproachesinfinity

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

13. xapproachesinfinity

it is not quite zero but not 1 either!

14. Michele_Laino

no, there is not a logarithm of negative number

15. xapproachesinfinity

you mean log base negative ?

16. mathmath333

17. mathmath333

i mean where x can be taken negative

18. xapproachesinfinity

no cannot x>0 always

19. Michele_Laino

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

20. xapproachesinfinity

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

21. xapproachesinfinity

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

22. Michele_Laino

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

23. Michele_Laino

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

24. xapproachesinfinity

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

25. Michele_Laino

that's right! @xapproachesinfinity