Why do the properties of logarithms change the bounds of this function? I started with the equation:
$$
f(x)=5\log_2 (x^3 +1)+\log_2 (x)-\log_2 (3)
$$
This gave me this graph:

Using the properties of logarithms, I was able to simplify the equation to this (which should be equivalent).
$$
y=\log_{2}\left(\frac{\left(x^{3}+1\right)^{5}\left(x\right)}{3}\right)
$$
I looked at the graph and got this:

This graph was the same as the first one with the positive X values, but it also had negative X values as well. This confused be because the two equations are apparently equivalent (as confirmed with some websites). Why do they have different graphs?
 A: the equality $log(ab)=log(a)+log(b)$ is true provided that $a,b>0$,otherwise the RHS doesn't make sense,for example $log(-7\times -7)$ is defined ,whereas $log(-7)+log(-7)$ doesn't make sens.Now two functions $f,g$ are equal iff they have the same domain of definition $D$ and $\forall x\in D , f(x)=g(x)$ but this is not the case in your example,the first function is defined for $x\in \left ] 0;+\infty \right [$,whereas the second one is defined whenever $(x^{3}+1)x>0$ ,that is when $x\in \left] -\infty ;-1 \right [ \cup \left ] 0;+\infty \right [$ .
A: The grapher you are using restricts the domain of a function based on your input.
In the first case, the domain is restricted to positive values of x, due to the logarithm of x. The grapher assumes f(x) as a sum of functions, and then consider each of their domains.
However, in the case of y, there is only one function, and therefore only one domain to be considered which is broader than the domain for f(x). Y has some values defined for x<0 while f(x) has none, which is why the graph of y also extends to the left.
