SOLUTION: If 2 log (base 3) y - 3 log (base 3) x - log (base 3) b + log (base 3) a = 3, express y in terms of a. b and x. Help Please!

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: If 2 log (base 3) y - 3 log (base 3) x - log (base 3) b + log (base 3) a = 3, express y in terms of a. b and x. Help Please!      Log On


   

Question 824073: If 2 log (base 3) y - 3 log (base 3) x - log (base 3) b + log (base 3) a = 3,
express y in terms of a. b and x.
Help Please!
Found 2 solutions by stanbon, jsmallt9:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
If 2 log (base 3) y - 3 log (base 3) x - log (base 3) b + log (base 3) a = 3,
--------
log3(y^2) - log3(x^3) - log3(b) + log3(a) = 3
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log3[y^2*a/(x^3*b)] = 3
----------------------------
(y^2*a)/(x^3*b) = 27
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y^2 = (27x^3b)/a
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y = [27x^3b/a]^(1/2)
===========================
y = (3x)sqrt(3xb/a)
===========================
Cheers,
Stan H.
================

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!

First we will combine the logs using the following properties:
  • log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29
  • log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29
  • n%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5En%29%29
The first two will be used to combine the logs. But they require that the logs have the same base and coefficients of 1. The third property is used to "move" pesky coefficients out of the way so the other two properties may then be used.

So we start with the third property:

Now we can start combining the logs. Because of the "-" between the first two logs, we will use the second property to combine them:
log%283%2C+%28y%5E2%2Fx%5E3%29%29+-+log%283%2C+%28b%29%29+%2B+log%283%2C+%28a%29%29+=+3
The first two logs of what remains have a "-" between them so back to the second property:
log%283%2C+%28%28y%5E2%2Fx%5E3%29%2Fb%29%29+%2B+log%283%2C+%28a%29%29+=+3
which simplifies to:
log%283%2C+%28y%5E2%2F%28b%2Ax%5E3%29%29%29+%2B+log%283%2C+%28a%29%29+=+3
Next we use the first property to combine the remaining logs because they have a "+" between them:
log%283%2C+%28%28y%5E2%2F%28b%2Ax%5E3%29%29%2Aa%29%29+=+3
which simplifies to:
log%283%2C+%28%28a%2Ay%5E2%29%2F%28b%2Ax%5E3%29%29%29+=+3

Now that the logs are all combined, we will rewrite the equation in exponential form. In general log%28a%2C+%28p%29%29+=+n is equivalent to p+=+a%5En. Using this pattern on our equation we get:
%28a%2Ay%5E2%29%2F%28b%2Ax%5E3%29+=+3%5E3
which simplifies to:
%28a%2Ay%5E2%29%2F%28b%2Ax%5E3%29+=+27

Now that the logs are gone, we can solve for y. Multiplying each side by b%2Ax%5E3 (to eliminate the fraction):
a%2Ay%5E2+=+27b%2Ax%5E3
Dividing by a:
y%5E2+=+%2827b%2Ax%5E3%29%2Fa
And finally, a square root of each side. (Note: Usually when one finds a square root of each side, a + is required to include both the positive and negative square roots. But in this problem the original equation had log%283%2C+%28y%29%29 in it. Because arguments of logs may never be negative we can "forget" the + on purpose since the negative square root is not a possibility.)
y+=+sqrt%28%2827b%2Ax%5E3%29%2Fa%29
Here's y expressed in terms of a, b and x.