using argmax in maximum a posteriori estimation from max likelihood estimation I was going through maximum a posteriori estimation and could not understand, why argmax is used for first term in image shown below but missing from second term. Shouldn't we have used argmax for both of them? If not what is the reasoning behind this.


You are misreading the notation.
$$
\underset{\theta}{\operatorname{arg\,max}} \log P(X|\theta) + \log P(\theta) = \underset{\theta}{\operatorname{arg\,max}} \Big( \log P(X|\theta) + \log P(\theta) \Big)
$$
