Mathematics Question and Solution
Calculating x.
Dividing through by 9^(x).
1+(21^(x))/9^(x)) = (49^(x)/9^(x))
1+(7/3)^(x) = (7/3)^(2x)
Let (7/3)^(x) = p
Therefore;
1+p = p²
Calculating p.
p²-p-1 = 0
(p-½)² = 1+(½)²
(p-½)² = (5/4)
p-½ = ±½√(5)
p = ½±½√(5)
It implies;
p = ½-½√(5)
Or
p = ½+½√(5)
Recall.
p = (7/3)^(x)
Therefore, calculating x.
For p = ½-½√(5)
½-½√(5) = (7/3)^(x)
Taking log base 10 at both sides of the equality.
log(½-½√(5)) = xlog(7/3)
x ≠ log(½-½√(5))/log(7/3)
For p = ½+½√(5)
½+½√(5) = (7/3)^(x)
Taking log base 10 at both sides of the equality.
log(½+½√(5)) = xlog(7/3)
x = log(½+½√(5))/log(7/3)
x = 0.56793702375
Checking accuracy.
9^(0.56793702375)+21^(0.56793702375) = 49^(0.56793702375)
Therefore;
9.11853209906 = 9.11853209904
It implies, our answer, x = 0.56793702375 is correct.
