Here are four identities that are true when the variable-domains are sufficiently restricted; but Mathematica doesn't evaluate them to "True", because they are not true when some variables take on complex or negative values:

In[1]:= a^x*b^x == (a*b)^x

Out[1]= a^x b^x == (a b)^x

In[2]:= (a^x)^y == a^(x*y)

Out[2]= (a^x)^y == a^(x y)

In[3]:= n*Log[x] == Log[x^n]

Out[3]= n Log[x] == Log[x^n]

In[4]:= Log[x] + Log[y] == Log[x*y]

Out[4]= Log[x] + Log[y] == Log[x y]

How can I ask Mathematica to tell me the exact conditions which make the identities valid? For which values are the identities true?

(I could solve this problem with enough thought or research, but I'm still curious how to ask Mathematica, in case I want to make similar queries about less common identities.)

Thanks.