Why does af(x) cause vertical dilation, while f(bx) causes horizontal dilation, when both of the increase/decrease the y value of any given x value?

Question

Especially for something like a log or a quadratic graph, where increasing “b” has virtually the same effect as increasing “a”. Both give a higher y-value for any given x-value, which should be considered vertical stretching (as it increases the gradient). And yet, when b is increased, f(bx) is considered as horizontal compression. Why is this?