Exponent Law of Addition? Is there a rule for adding exponential terms of like bases just like there are rules for multiplying and dividing such terms?
For example we know that:
$x^1 \cdot x^2 = x^{1+2} = x^3$
But what about for addition (or subtract for that matter)?
$x^1 + x^2 = x^?$
If no such pattern exists, why is that?

Observe that
$ \log(A+B) = \log(A) + \log \left(\frac{B}{A} + 1 \right) $. Using this fact, one can show that 
$
\begin{align}
\log (x^a + x^b) &= \log(x^a) + \log ( x^{b-a} + 1 ) \\
&= a\log x + \log ( x^{b-a} + 1 ) \\
&= \log x \left( a + \frac{\log ( x^{b-a} + 1 )}{\log x} \right) \\
\Rightarrow x^a + x^b &= x^{\left( a + \frac{\log ( x^{b-a} + 1 )}{\log x} \right)} \\
\text{In your case, with  }& a=1, b=2, \text{ we get} \\
x^1 + x^2 &= x^{\left( 1 + \frac{\log ( x + 1 )}{\log x} \right)}
\end{align}
$
