SOLUTION: 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.

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!

Square both sides

Divide both sides by

Solve for Have to run.
Hope I got it.

Answer by Theo(13342) (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.

RELATED QUESTIONS
I have two questions that I am confused on Please help! 1. Solve for x: log10 9 =... (answered by ankor@dixie-net.com)
log10 x * log10... (answered by Alan3354)
I need help with this problem. Directions is solve for y in terms of x. log10y = log10... (answered by Alan3354)
Find the unknown quantity in the given equation. log10 x - 2 log10 3= log10... (answered by MathLover1)
How do I solve: log10(x+8)+log10(x-8)? Simplifying the logarithm into 1 log. Would (answered by stanbon)
Logx+log(x+3)=log10 (answered by lwsshak3)
given that log10^7=x and log10^2=y, evaluate... (answered by Theo,Edwin McCravy)
Okay, I am so lost. Please help me with these? Write as the sum or difference of a... (answered by Nate)
1) Solve for x: 2-x = 1 / 32 2) Solve for x: 34x + 5 = 9 (answered by stanbon)