Solve logarithmic equations Set 2

Learning Objectives Apply the product, quotient, and power rules to condense logarithmic expressions within an equation. Solve logarithmic equations by condensing terms and converting the equation to exponential form. Solve logarithmic equations by condensing terms on both sides and applying the one-to-one property. Identify and solve quadratic equations that arise from solving logarithmic equations. Check all potential solutions to identify and discard extraneous solutions. Interpret the domain restrictions of logarithmic functions in the context of equation solving. How can we solve for the intensity of an earthquake if it's hidden inside a logarithm in the Richter scale formula? 🌋 Let's learn the techniques to unlock those variables! This tutorial builds on yo...

TermDefinitionExample Logarithmic EquationAn equation that includes a variable within the argument of a logarithm.log₃(x + 5) = 4 Condensing LogarithmsThe process of using logarithm properties (Product, Quotient, Power Rules) to combine multiple logarithmic terms into a single logarithm.log(x) + log(2) is condensed to log(2x). One-to-One Property of LogarithmsIf two logarithms with the same base are equal, then their arguments must also be equal.If log₅(x - 1) = log₅(10), then x - 1 = 10. Extraneous SolutionA solution that is correctly derived from the algebraic steps of solving an equation, but is not a valid solution to the original equation because it violates a domain restriction.In log(x) + log(x - 3) = 1, the potential solution x = -2 is extraneous because log(-2) is undefined. Doma...

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 into 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 that are being subtracted into 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 into the argument as an exponent.