Question 658889: knowing that log10((x+y)/3)=1/2(log10(x)+log10(y)) determine the value for x/y
P.S - the 3 is inside the log and the log base is 10. Found 2 solutions by josmiceli, Theo:Answer by josmiceli(19441) (Show Source):
You can put this solution on YOUR website! here's my take on solving this problem.
presumably you want to solve for x/y
since log10(x) is the same as log(x), i'll rewrite the problem as follows:
log((x+y)/3) = 1/2 * (log(x) + log(y)
since log(a) + log(b) is equal to log(ab), the equation becomes:
log((x+y)/3) = 1/2 * log(xy)
since a * log(b) = log(b^a), the equation becomes:
log((x+y)/3) = log((xy)^(1/2)) which is the same as:
log((x+y)/3) = log(sqrt(xy))
since log (a/b) = log(a) - log(b), the equation becomes:
log(x+y) - log(3) = log(sqrt(xy))
add log(3) to both sides of the equation and subtract log(sqrt(xy)) from both sides of the equation to get:
log(x+y) - log(sqrt(xy)) = log(3)
since log(a) - log(b) = log(a/b), the equation becomes:
log((x+y)/sqrt(xy)) = log(3)
if log(a) = log(b) then a must be equal to b so the equation becomes:
(x+y)/sqrt(xy)) = 3
multiply both sides of this equation by sqrt(xy) to get:
x+y = 3*sqrt(xy)
divide both sides of this equation by y to get:
(x+y)/y = 3*sqrt(xy)/y
simplify to get:
x/y + 1 = 3*sqrt(xy)/y
subtract 1 from both sides of this equation to get:
x/y = 3*sqrt(xy)/y - 1
if i assumed a value 1 for y then i was able to solve for x to get:
x = 6.854101966 and x = .1458980338
i confirmed that when x = 6.85... and y = 1, the original equation is true.
i also confirmed that when x = .14... and y = 1, the original equation is also true.
i'm thinking this confirms the solution is good.
hopefully you and/or your instructor agree.
good luck.
interesting problem.
