Use the change of base formula, log_a(n) = logn/loga, to start the solving process.
log(x + 6)/log3 - logx/log9 = log2/log9
log(x + 6)/log3 - logx/log3^2= log2/log3^2
log(x + 6)/log3 - logx/(2log3) = log2/(2log3)
Put on a common denominator.
(2log(x + 6))/(2log3) - logx/(2log3) = log2/(2log3)
log_9(x + 6)^2 - log_9x = log_9 2
We now use the property log_a(n) - log_a(m) = log_a(n/m) to solve.
log_9((x + 6)^2/x) = log_9 2
(x^2 + 12x + 36)/x = 2
x^2 + 12x + 36 = 2x
x^2 + 10x + 36 = 0
x = (-b +- sqrt(b^2 -4ac))/(2a)
x = (-10 +- sqrt(10^2 - (4 xx 1 xx 36)))/(2(1))
x = (-10 +- sqrt(-44))/2
x = (-10 +- 2sqrt(11)i)/2
x = -5 +- sqrt(11)i
This does not satisfy the original equation, and hence there are no solutions.
Hopefully this helps!
