How do you graph 2log(base5)(x-5)-log(base5)(x^2-25)^2+log(base5)(x^2+5x+20)^2+2?

Question

I need to know how to graph this logarithmic equation and have no idea how to. Please help!

Answer 1

This would greatly simplify if x²+9x+20 replaced x²+5x+20, because many terms would cancel out. Check your original question.

x²+9x+20=(x+4)(x+5) so the expression simplifies:

2log₅(x-5)-2log₅(x²-25)+2log₅(x²+9x+20)+2log₅(5)=

2log₅{5(x+5)(x+4)(x-5)/[(x-5)(x+5)]}=2log₅[5(x+4)].

Note that x=-4 is a vertical asymptote.

When x=-3.8, x+4=0.2=⅕, so log₅[5×⅕]=log₅(1)=0 so x=-3.8 is the x-intercept.

The points (-3,2) and (1,4) are also on the graph, and these points as well as the asymptote enable you to sketch the graph.