I want to confirm the domain and range for the composite function: f(g(x)) and g(f(x)) for the functions:

f(x)=10^x and g(x)=log x

For f(g(x)) the domain I got is x>0 and the range is y>0. And for g(f(x)) the domain and range is all real numbers.

f(g(x)) = 10^(logx). Log(x) can't be negative, so the domain and range is anything above 0.
g(f(x)) = log(10^x). 10^x will always be positive, and the log of something positive has no restraints, so it's all real numbers.

So you're right :D Yay!