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(35256)   (Show Source): You can put this solution on YOUR website!
Start with the given equation


Combine the logs


Use the relationship <===> to rewrite the equation


Square 5


Multiply both sides by


Distribute


Get all terms to one side


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


Starting with the general quadratic



the general solution using the quadratic equation is:





So lets solve ( notice , , and )




Plug in a=1, b=-25, and c=-50



Negate -25 to get 25



Square -25 to get 625 (note: remember when you square -25, you must square the negative as well. This is because .)



Multiply to get



Combine like terms in the radicand (everything under the square root)



Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

or




So these expressions approximate to

or



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

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