Logistic regression cost function In Aurelien Geron's  book I found this line

This cost function makes sense because –log(t) grows very large when t approaches
 0, so the cost will be large if the model estimates a probability close to 0 for a positive instance, and it will also be very large if the model estimates a probability close to 1
for a negative instance. On the other hand, – log(t) is close to 0 when t is close to 1, so
the cost will be close to 0 if the estimated probability is close to 0 for a negative
instance or close to 1 for a positive instance, which is precisely what we want.

What I dont get is, How will  the cost will be large if the model estimates a probability close to 0 for a positive instance, and it will also be very large if the model estimates a probability close to 1
for a negative instance?
The cost function of the Logistic Regression derived via Maximum    Likelihood  Estimation: 




If y = 1 (positive): i) cost = 0 if prediction is correct (i.e. h=1), ii) cost $\rightarrow \infty $ if $h_{\theta}(x)\rightarrow 0$.
If y = 0 (negative): i) cost = 0 if prediction is correct (i.e. h=0), ii) cost $\rightarrow \infty$ if $(1-h_{\theta}(x))\rightarrow 0$.


The intuition is that larger mistakes should get larger penalties.
Further readings, 1,2,3,4.