+170 There are several real numbers \(x\) such that \(log_2 (x + 2), \ \log_4 (3x + 4), \ \log_8 (7x + 8)\)
are three real numbers in arithmetic progression in the order listed. One such number \(x\) can be expressed in the form \(\frac{-a + b \sqrt{c}}{d},\)
where \(a,b,c,d\) are positive integers. Find \(a+b+c+d\)(It is assumed that the number in this form is as simplified as possible.)
+129899 Using the change-of base theorem we have
log (3x + 4) log (3x + 4) log (3x + 4) (1/2)log(3x + 4)
_________ = __________ = ______________= ____________ (1)
log 4 log 2^2 2 log 2 log 2
log ( 7x + 8) log(7x + 8) log (7x + 8) (1/3)log(7x + 8)
_________ = __________ = _____________ = _____________ (2)
log 8 log 2^3 3 log 2 log 2
The arithmetic difference must be (2) - (1) =
(1/3) log (7x + 8) - (1/2)log(3x + 4)
__________________________
log 2
This implies that
log (x + 2) (1/3)log(7x + 8) - (1/2)log (3x + 4) (1/2)log(3x + 4)
________ + ____________________________ = ______________
log 2 log 2 log 2
;log (x + 2) + (1/3)log(7x + 8) - (1/2)log(3x + 4) = (1/2)log(3x + 4)
log (x + 2) + (1/3)log(7x + 8) = log(3x + 4)
3 log( x + 2) + log(7x + 8) = 3log(3x + 4)
log ( x + 2)^3 + log(7x + 8) = log (3x + 4)^3
log [ (x + 2)^3 * (7x + 8)] = log(3x + 4)^3
Which implies that
(x + 2)^3 (7x + 8) = (3x + 4)^3
(x^3 + 3x^2*2 + 3x*4 + 2^3) (7x + 8) = (3x)^3 + 3* (3x)^2*4 + 3* 3x*4^2 + 4^3
( x^3 + 6x^2 + 12x + 8)(7x + 8) = 27x^3 + 108x^2 + 144x + 64
7 x^4 + 50 x^3 + 132 x^2 + 152 x + 64 = 27x^3 + 108x^2 + 144x + 64
7x^4 + 23x^3 + 24x^2 + 8x = 0
x (7x^3 + 23x^2 + 24x + 8) = 0
x = 0 is one solution
7x^3 + 23x^2 + 24x + 8 = 0
Using the Rational zeroes theorem, I see that x = -1 is also another solution
So....we can use synthetic division to find the residual polynomial
-1 [ 7 23 24 8 ]
-7 -16 -8
________________
7 16 8 0
The remaining polynomial is 7x^2 + 16x + 8
Using the Quadratic Formula
x = -16 ±√ [ 16^2 - 4(7)(8) ] = -16 ±√ 32 = -16 ± 4√2 = -8 ±2√2
____________________ __________ _________ _______
2 *7 14 14 7
The solution -8 - 2√2
_______ ≈ -1.55 which makes two of the original logs undefined
7
So
x = -8 + 2 √2
_________ ≈ -.74
7
a = 8 b = 2 c = 2 d = 7
And their sum = 19
