6 = 2 • 3, so
6ˣ⁻³ = 2ˣ⁻³ • 3ˣ⁻³
Then in
2²ˣ = 6ˣ⁻³
we would have
2²ˣ = 2ˣ⁻³ • 3ˣ⁻³
2²ˣ / 2ˣ⁻³ = 3ˣ⁻³
2²ˣ⁻ˣ⁺³ = 3ˣ⁻³
2ˣ⁺³ = 3ˣ⁻³
Now take the logarithm of both sides. The base you choose doesn't matter.
log(2ˣ⁺³) = log(3ˣ⁻³)
Recall that log(aⁿ) = n log(a), so that the equation reduces to
(x + 3) log(2) = (x - 3) log(3)
Solve for x :
log(2) x + 3 log(2) = log(3) x - 3 log(3)
(log(2) - log(3)) x = - 3 log(2) - 3 log(3)
Recall that log(a) - log(b) = log(a/b), and log(a) + log(b) = log(ab), so that
log(2) - log(3) = log(2/3)
- 3 log(2) - 3 log(3) = - 3 (log(2) + log(3)) = - 3 log(6)
So, we get
log(2/3) x = - 3 log(6)
x = -3 log(6) / log(2/3)
As I mentioned, the base of the logarithm doesn't matter, x will always have the same value of about 13.257.
