Step-by-step explanation:
log10(18) = log10(2×3×3) = a
log10(24) = log10(2×2×2×3) = b
the rules of the logarithm :
logc(m×n) = logc(m) + logc(n)
log10(18) = log10(2) + log10(3) + log10(3) = a
log10(2) + 2×log10(3) = a
log10(24) = log10(2) + log10(2) + log10(2) + log10(3) = b
3×log10(2) + log10(3) = b
log10(2) = a - 2×log10(3)
using this in the b equation :
3×(a - 2×log10(3)) + log10(3) = b
3a - 6×log10(3) + log10(3) = b
3a - 5×log10(3) = b
-5×log10(3) = b - 3a
log10(3) = -(b - 3a)/5 = (3a - b)/5
similarly, from the b equation we get :
log10(3) = b - 3×log10(2)
and we use that in the original a equation :
log10(2) + 2×(b - 3×log10(2)) = a
log10(2) + 2b - 6×log10(2) = a
2b - 5×log10(2) = a
-5×log10(2) = a - 2b
log10(2) = -(a - 2b)/5 = (2b - a)/5
