SOLUTION: solve for x: log base 3 (7x+4)-log base 3 of 2 = 2logbase3 of X

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Question 560460: solve for x:
log base 3 (7x+4)-log base 3 of 2 = 2logbase3 of X
Found 2 solutions by stanbon, josmiceli:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
solve for x:
log base 3 (7x+4)-log base 3 of 2 = 2logbase3 of X
----
log3(7x+4) - log3(2) = log3(x^2)
-------
log3[(7x+4)/2] = log3(x^2)
------
[(7x+4)/2] = x^2
----
2x^2 - 7x - 2 = 0
---
x = [7 +- sqrt(49-4*2*-2)]/4
-------
x = [7 +- sqrt(49+16)]/4
----
Positive solution:
-----
x = [7 + sqrt(65)]/4
----
x = 3.7656..
================
Cheers,
Stan H.

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+log%283%2C+%287x+%2B+4%29%29+-+log%283%2C2%29+=+2%2Alog%283%2Cx%29+
Use the rule
+log%28a%29+-+log%28b%29+=+log%28a%2Fb%29+
+log%283%2C++%28%287%2F2%29%2Ax+%2B+2%29+%29+=+2%2Alog%283%2Cx%29+
Use the rule
+a%2Alog%28b%29+=+log%28b%5Ea%29+
+log%283%2C+%28%287%2F2%29%2Ax+%2B+2%29+%29+=+log%283%2Cx%5E2%29+
+%287%2F2%29%2Ax+%2B+2+=+x%5E2+
+x%5E2+-+%287%2F2%29%2Ax+-+2+=+0+
+2x%5E2+-+7x+-+4+=+0+
Use the quadratic formula
+x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
+a+=+2+
+b+=+-7+
+c+=+-4+
+x+=+%28-%28-7%29+%2B-+sqrt%28+%28-7%29%5E2-4%2A2%2A%28-4%29+%29%29%2F%282%2A2%29+
+x+=+%28+7+%2B-+sqrt%28+49+%2B+32+%29%29+%2F+4+
+x+=+%28+7+%2B-+sqrt%28+81+%29%29+%2F+4+
+x+=+%28+7+%2B+9+%29%2F4+
+x+=+4+
and, taking the (-) square root,
+x+=+%28+7+-+9+%29+%2F+4+
+x+=+-2%2F4+
+x+=+-%281%2F2%29+
--------------
Using +x+=+4+,
+log%283%2C+%287x+%2B+4%29%29+-+log%283%2C2%29+=+2%2Alog%283%2Cx%29+
+log%283%2C+%287%2A4+%2B+4%29%29+-+log%283%2C2%29+=+2%2Alog%283%2C4%29+
+log%283%2C+32%29+b-+log%283%2C+2%29+=+2%2Alog%283%2C4%29+
+log%283%2C+%2832%2F2%29%29+=+log%283%2C4%5E2%29+
+log%283%2C16%29+=+log%283%2C16%29+
OK
Using +x+=+-%281%2F2%29+

I have to reject this solution for x because of the
right side. It is impossible to raise the base 3 to
some power ( + or - ) that will result in +-%281%2F2%29+,
or any negative number.
So, +x+=+4+ is the only solution