In trigonometry, there is a relation called the addition theorem:
$\begin{align} \sin(A+B) &= \sin{A}\cos{B}+\cos{A}\sin{B} \\ \cos{(A+B)} &= \cos{A}\cos{B} -\sin{A}\sin{B}\end{align}$
I wondered if there is a similar relation for logarithms. So what I want to know is that, is there a polynomial or a rational function $f(x,y)$ that satisfies $\log(a+b)= f(\log(a),\log(b))$?(Assume $a>0,b>0$)
I think it is highly unlikely, but I haven't been able to prove that. I also would love to know if we can loosen the restriction on $f(x,y)$ and make it true.
No, it's the other way round:
$$\exp(a+b)=\exp(a)\exp(b)$$
or
$$\log(ab)=\log(a)+\log(b).$$
In fact, via the complex numbers, the first identity directly leads to the trigonometric ones.
We can refute the hypothesis $$\log(a+b)=f(\log(a)+\log(b))$$ for rational functions with algebraic coefficients.
Indeed,
$$\log(2)=\log(1+1)=f(\log(1),\log(1))=f(0,0)$$ is a transcendental number, but cannot be the result of a rational function with algebraic coefficients applied to algebraic arguments.