SOLUTION: Find the largest value of x that satisfies: log(base5) (x^2) − log(base5) (x+2) = 2 x = THANKS A LOT!

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Question 139457: Find the largest value of x that satisfies:
log(base5) (x^2) − log(base5) (x+2) = 2
x =
THANKS A LOT!
Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!
log%285%2C%28x%5E2%29%29-log%285%2C%28x%2B2%29%29+=+2+ Start with the given equation


log%285%2Cx%5E2%2F%28x%2B2%29%29+=+2+ Combine the logs


x%5E2%2F%28x%2B2%29+=+5%5E2+ Use the relationship log%28b%2C%28y%29%29=x <===> b%5Ex=y to rewrite the equation


x%5E2%2F%28x%2B2%29+=+25+ Square 5


x%5E2+=+25%28x%2B2%29+ Multiply both sides by x%2B2


x%5E2+=+25x%2B50+ Distribute


x%5E2-25x-50=0++ Get all terms to one side


Let's use the quadratic formula to solve for x:


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve x%5E2-25%2Ax-50=0 ( notice a=1, b=-25, and c=-50)




x+=+%28--25+%2B-+sqrt%28+%28-25%29%5E2-4%2A1%2A-50+%29%29%2F%282%2A1%29 Plug in a=1, b=-25, and c=-50



x+=+%2825+%2B-+sqrt%28+%28-25%29%5E2-4%2A1%2A-50+%29%29%2F%282%2A1%29 Negate -25 to get 25



x+=+%2825+%2B-+sqrt%28+625-4%2A1%2A-50+%29%29%2F%282%2A1%29 Square -25 to get 625 (note: remember when you square -25, you must square the negative as well. This is because %28-25%29%5E2=-25%2A-25=625.)



x+=+%2825+%2B-+sqrt%28+625%2B200+%29%29%2F%282%2A1%29 Multiply -4%2A-50%2A1 to get 200



x+=+%2825+%2B-+sqrt%28+825+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%2825+%2B-+5%2Asqrt%2833%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%2825+%2B-+5%2Asqrt%2833%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%2825+%2B+5%2Asqrt%2833%29%29%2F2 or x+=+%2825+-+5%2Asqrt%2833%29%29%2F2




So these expressions approximate to

x=26.861 or x=-1.861



So we can clearly see that x=26.861 is the largest value of x that satisfies the equation