is log 0 is or is not undefined? per Figure 1.7 in pattern recognition and machine learning (free)

Plots of M = 9 polynomials fitted to the data set shown in Figure 1.2 using the regularized error
  function (1.4) for two values of the regularization parameter λ corresponding to ln λ = −18 and ln λ = 0. The
  case of no regularizer, i.e., λ = 0, corresponding to ln $λ = −∞$

ln $0 = −∞$
per this post

log 0 is undefined.

so, is log 0 is or is not undefined?


The logarithmic function $~\log_b(x)~$ (for any positive real number $b\ne 1$) is defined only for $~x>0~$.

If possible let $~\log_b(0)=k~$, where $~k~$ is any real number. Then by the definition of logarithm, $~b^k=0~$. 

Now, any ‘real’ quantity (may it be positive or negative) raised to another real quantity can never result in $~0~$. 

You may think about the case of $(\text{any number})^{0}$ or more specifically, $0^0$. But the interesting fact is that any number raised to the power $0$ results in $1$ and not $0$.
So, the value of the expression $~\log_b(0)~$ makes no sense and more specifically, we cannot determine any value for it or we cannot define it.

Thus $~\log_b(0)~$ is Indeterminate or Undefined or Does not exist.


For various values of the base $~b~$ $($let us consider $~b=10,~b=e,~b=2$ $)$, at $~x=0$, we have vertical asymptotes (see in the figure given below). 
 
