Logarithmic equations
Equations and inequalities · Mathematics
Study notes
Q: Solve (a) log₂(x) + log₂(x - 2) = 3 (b) log(x + 1) - log(x) = 1 (base 10). (a) Combine: log₂(x(x-2)) = 3. Exponentiate: x(x - 2) = 8. x² - 2x - 8 = 0: (x-4)(x+2) = 0. x = 4 or -2. Check: x = -2 gives log₂(-2), invalid. x = 4: log₂4 + log₂2 = 2 + 1 = 3 ✓. Answer: 4. (b) Combine: log((x+1)/x) = 1. So (x+1)/x = 10. x + 1 = 10x, 9x = 1, x = 1/9.