Solve logarithmic equations with one logarithm

Learning Objectives Convert a logarithmic equation with a single logarithm into its equivalent exponential form. Isolate the logarithmic term using algebraic principles before solving. Solve for the variable in linear and quadratic arguments of logarithmic equations. Determine the domain of a logarithmic equation to identify potential restrictions on the variable. Verify solutions and correctly identify and discard extraneous solutions. Solve equations involving common logarithms (base 10) and natural logarithms (base e). How do scientists calculate the age of a fossil or the intensity of an earthquake? They use logarithmic equations to model these powerful real-world phenomena! 🦴️ This tutorial focuses on the fundamental skill of solving equations containing a single loga...

TermDefinitionExample Logarithmic EquationAn equation that contains a variable within the argument of a logarithm.log₃(x + 5) = 2 is a logarithmic equation. ArgumentThe expression inside the logarithm. The argument of a logarithm must always be a positive number.In the expression log₅(2x - 3), the argument is (2x - 3). Domain of a LogarithmThe set of all possible input values (for the variable) that keep the argument positive.For log(x - 4), the argument (x - 4) must be greater than 0. Therefore, the domain is x > 4. Exponential FormThe equivalent representation of a logarithmic relationship. The equation log_b(y) = x is equivalent to b^x = y.The logarithmic equation log₂(8) = 3 can be rewritten in exponential form as 2³ = 8. Extraneous SolutionA result obtained through correct algebra...

Logarithm-to-Exponential Conversion log_b(A) = c <=> b^c = A This is the fundamental conversion used to solve logarithmic equations. Once the logarithm is isolated, rewrite the equation in exponential form to eliminate the log and solve for the variable in the argument 'A'. The Isolation Principle If k * log_b(A) + m = n, first rewrite as log_b(A) = (n - m) / k Before converting to exponential form, you must isolate the logarithmic term. Use standard inverse operations (addition/subtraction, then multiplication/division) to get the log expression by itself on one side of the equation.