I'm very new to math, and I wanted to go back to basics and learn everything on my own. I know that $\log{(xy)}=\log{(x)}+\log{(y)}$ for all real numbers. The rationale is easy to explain for integers. Say your logarithm is in base $2$. And say you're adding $\log_2(4)$ and $\log_2$(8). $\log_2(4)=2$, because $2^2$ is $4$. $\log_2$(8) is $3$, because $2^3=8$. Now if we multiply $2^2\cdot 2^3$ and expand, it's $(2\cdot 2)\cdot (2\cdot 2\cdot 2)$. The number of "twos" here are $2+3$, so it's $2^{(2+3)}$. However, it's harder to prove the same for fractional exponents (in whatever form, improper or proper). The only thing I can sus out is that all can be rewritten in exponential form. Example, $11^{3/4} $ is $(\sqrt[3]{11})^4$, and $12^{1/2}$=$\sqrt{12}^{1}$.
I can't figure this out. If someone can help, it would be great. Thank you.
To further iterate, I should've asked to prove the index rule of creating a number with a fractional exponent, and to prove it follows all the same rules integer exponents do.