2.2.3—Log equations

To solve a logarithmic equation, combine logs only when arguments are positive, then exponentiate or change base. Every candidate must satisfy the original log domains.

If log terms have different bases, convert consistently. Squaring or exponentiating can create algebraic candidates that the original equation rejects.

ln(x−1)+ln(x+1)=ln3 becomes ln(x²−1)=ln3 with x>1, so x=2; x=−2 is rejected by the domain.

Equality of log expressions does not allow non-positive arguments, even if a later algebraic step produces a real number.