Tristin S. answered • 03/08/21
Recent College Graduate Looking for Opportunities to Tutor Others
Since both logs are the same base we can do an old log trick. loga(x) - loga(y) = loga(x/y). In this case, since we have log2(x2) - log2(x+5) = 6, this becomes log2(x2/(x+5)) = 6.
In order to get rid of the log, we just take 2, and raise it to both sides or in math notation:
2 ^ log2(x2/(x+5)) = 26 (the carat (^) and superscript mean the same thing, it's just difficult to format)
x2/(x+5) = 26
x2/(x+5) = 64
x2 = 64(x+5) (this is ok, provided we don't have x = -5 as a root )
x2 = 64x + 320
x2 - 64x - 320 = 0
This equation doesn't have any recognizable roots that I can see, so we can use the quadratic formula to solve for its roots:
x = -(-64)± √(-64)2 - 4(1)(-320)/2
x = 64 ± √(-64)2 + 4(1)(-320) / 2
x = 64 ± √(4096+ 1280) / 2
x = 64 ± √5376 /2
x = 32 ± √1344 (divided out by √4 or 2)
Since only one of these solutions (x = 32 + √1344) is positive, that is the solution to our equation.
