Solve logarithmic equations Set 1

Learning Objectives Convert a logarithmic equation into its equivalent exponential form. Solve basic logarithmic equations of the form log_b(x) = c. Apply the one-to-one property to solve equations of the form log_b(x) = log_b(y). Isolate a single logarithmic term before solving the equation. Identify and discard extraneous solutions by checking the domain of the original equation. How can we find the exact time it takes for a city's population to double, or measure the intensity of an earthquake? 📈 The key is hidden inside logarithmic equations! This tutorial introduces the fundamental techniques for solving logarithmic equations. You will learn how to rewrite equations into a more familiar form and use key properties to find the value of the unknown variable. Masterin...

TermDefinitionExample Logarithmic EquationAn equation that includes a logarithm involving a variable.log₂(x + 3) = 4 is a logarithmic equation. Argument of a LogarithmThe expression inside the logarithm. The argument must always be a positive number.In log₃(2x - 1), the argument is (2x - 1). Domain of a LogarithmThe set of all valid inputs (arguments) for a logarithm. For log_b(A), the domain requires that A > 0.For log₅(x - 4), the domain is x - 4 > 0, which means x > 4. Extraneous SolutionA result obtained from solving an equation that is not a valid solution to the original equation. In logarithms, this usually occurs when a solution makes an argument less than or equal to zero.If solving an equation yields x = -5, but the original equation was log₂(x), then -5 is an extraneou...

Definition of a Logarithm (Conversion Rule) log_b(A) = c <=> b^c = A This is the most fundamental rule for solving logarithmic equations. It allows you to rewrite a logarithmic equation as an exponential equation, which is often easier to solve. This is used when you have a single logarithm isolated on one side of the equation. One-to-One Property of Logarithms If log_b(A) = log_b(C), then A = C Use this property when you have a single logarithm with the same base on both sides of the equation. It allows you to drop the logarithms and set the arguments equal to each other.