So, I asked a question earlier that basically asked "prove that $log(xy)=log(x)+log(y)$ for decimal/fraction logarithms", but I had not explained what I meant correctly. I meant: how can one prove that the law that $a^b*a^c= a^{b+c}$ holds for all real number exponents? For integers, as I said in my last post, it's rather easy. Just expand them and you'll notice that it holds true. Example:
$10^3*10^3 = (10*10*10)*(10*10*10) = 10^6$
However, this cannot be done with fractional or decimal exponents, at least in the form they're in.
Example: $10^{0.4} * 10^{0.9} = 10^{0.4+0.9}$. This holds true, but why does it hold true?
Thank you.
If $a > 0$, then taking $\log_a$ of both sides gives:
$\log_a(a^b * a^c) = \log_a(a^{b+c})$
$\log_a(a^b) + \log_a(a^c) = {b+c}$
$b + c = {b+c}$
If you are still not convinced, then try evaluating for real numbers.
Evaluate $10^{0.4}$ and $10^{0.9}$ on a software and multiply them together. Then check if the output is equal to $10^{0.4+0.9}$.