Logarithmic equations and original domains

A real logarithmic equation is defined only when every original argument is positive and each base is positive and different from 1. Combining logarithms does not remove those original restrictions. For a valid base, equal logarithms have equal arguments: log_a M=log_a N implies M=N. This follows from the one-to-one exponential inverse. Method: write all original domain conditions; combine same-base logarithms; convert to an ordinary equation; solve; test every candidate against the original domain. For ln(x-1)+ln(x+1)=ln8, the original domain is x>1. Combining gives x2-1=8, hence x=+-3. Only x=3 is valid. Exam check: the product of two negative arguments may be positive. A positive product alone does not make the two original logarithms exist. For equations quadratic in a logarithm itself, use the separate reducible-equations skill.