`log 0 = x` can be rewritten as `10^x = 0`. Since no value of x solves this equation, the expression log(0) is undefined.
However, notice that `10^x` gets pretty close to zero if you use large negative values for x. In fact, you can get `10^x` to be as close to zero as you desire if you choose a low enough x value. For this reason we can say `lim_(x->0+) log x = -infty`
Logorithims can be expressed as exponents and vice versa. For example, a^b = c can be expressed as loga (c)=b so, 3^2=9 or log3 (9) = 2... Following this logic would lead us to, in the case of base 0, logb(0) =a or b^a=0. This equation has no solution since there is no number that can be raised to any power and equal 0 so therefore, it is undefined. Hope this helps :)