Solve logarithmic equations with multiple logarithms

Learning Objectives Apply the product, quotient, and power rules to condense multiple logarithmic terms into a single logarithm. Solve logarithmic equations by converting them into exponential form after condensation. Utilize the one-to-one property of logarithms to solve equations where both sides are single logarithmic expressions with the same base. Determine the domain of logarithmic equations and identify and discard extraneous solutions. Solve equations that result in linear or quadratic forms after the logarithmic properties are applied. Use the change of base formula to approach problems with logarithms of different bases. How can we combine the intensity of two separate earthquakes to find their total power on the Richter scale? seismograph️ The answer lies in mast...

TermDefinitionExample Logarithmic EquationAn equation that includes at least one term containing the logarithm of a variable.log₃(x) + log₃(x-8) = 2 Condensing LogarithmsThe process of using logarithm properties (Product, Quotient, Power Rules) to combine multiple logarithmic terms into a single logarithm.log(x) + log(5) can be condensed to log(5x). Domain of a LogarithmThe set of valid input values for the argument of a logarithm. The argument must always be a positive number.For log(x-4), the domain is x > 4, because x-4 must be greater than 0. Extraneous SolutionA solution that is correctly derived algebraically but is not a valid solution to the original equation because it falls outside the domain of one or more of the original logarithmic terms.Solving log₂(x) + log₂(x-2) = 3 yie...

Product Rule for Logarithms log_b(M) + log_b(N) = log_b(MN) Use this rule to combine two logarithms with the same base that are being added. The arguments M and N are multiplied within a single logarithm. Quotient Rule for Logarithms log_b(M) - log_b(N) = log_b(M/N) Use this rule to combine two logarithms with the same base where one is subtracted from the other. The arguments are divided within a single logarithm. Power Rule for Logarithms p * log_b(M) = log_b(M^p) Use this rule to move a coefficient in front of a logarithm to become an exponent on the argument inside the logarithm.