Why Do Logarithmic Graphs Vary Despite Similar Equations?

[SwAnK]

I graphed y=logx+log2x and y=log2x^2 and the graphs came out a lot different. However when you simplify y=logx+log2x you actually get y=log2x^2! Any ideas about why the graphs are so different yet the equations are pretty much the same? thanx

 

[dav2008]

Well notice that log(a) outputs real numbers only if a>0.

With that in mind, when x<0, in your first equation the inputs to the logs are negative so that function is not defined if you are graphing real numbers. The second equation is defined for x<0 because x² yields a positive value.

If you look at the two graphs they are identical when x>0. When x<0, the first graph is undefined while the second graph is just an even extension of when x>0.Just as a note, when you are dealing with the function y=log(x)+log(2x), the domain that yields a real result is x>0. If you simplify the expression to y=log(2x²) then you have to remember that your domain is still x>0. In that sense the two functions are identical.

 

[HallsofIvy]

In other words, log a+ log b= log ab if and only if log a and log b both exist- that is, as long as a and b are both positive. If a and b are both negative, log ab exists even though log a and log b do not.