if given log(a)=b, where no base is stated, we assume base 10, or [tex]log(a)=log_{10}(a)[/tex]
also, [tex]log_a(b)=c[/tex] translates to/can be written equivilently as [tex]a^c=b[/tex]
therefore, given
[tex]log(-13)=?[/tex] where we want to find ?
assume base 10 and translate
[tex]10^?=-13[/tex]
since we cannot raise a positive number to any real power and give a negative number, ? is not a real number and therefore, there are no real solutions
if we do want to find a solution, we can use Euler's idendity
[tex]e^{\pi i}=-1[/tex] where i is the complex number i=√-1 and e is euler's number
we can try to change bases to find the value of ?
[tex]e^{\pi i}=-1[/tex]
[tex]13e^{\pi i}=-13[/tex]
so
[tex]13e^{\pi i}=10^?[/tex]
taking ln of both sides
[tex]ln(13e^{\pi i})=ln(10^?)[/tex]
using log rules
[tex]ln(13)+(\pi i)ln(e)=(?)ln(10)[/tex]
[tex]ln(13)+\pi i=?ln(10)[/tex]
divie both side by ln(10)
[tex]\frac{ln(13)+\pi i}{ln(10)}=?[/tex]
in a+bi form
[tex]?=\frac{ln(13)}{ln(10)}+\frac{\pi}{ln(10)}i[/tex]
there are no real numbers that it evaluates to
however, it does evaluate to a complex number which is [tex]\frac{ln(13)}{ln(10)}+\frac{\pi}{ln(10)}i[/tex] or aproximately 1.11394+1.36438i
