Solve exponential equations using logarithms

Learning Objectives Identify when a logarithm is necessary to solve an exponential equation. Apply the Power Rule of logarithms to move a variable from an exponent into a coefficient. Solve exponential equations of the form a * b^(cx+d) = k. Solve exponential equations where variables appear in exponents on both sides of the equation. Use a calculator to find approximate decimal solutions for logarithmic expressions. Verify solutions by substituting them back into the original equation. How long would it take for an investment to triple if it earns 7% annual interest, compounded continuously? 🤔 Logarithms are the key to unlocking the answer! This tutorial focuses on the essential technique of using logarithms to solve exponential equations where the variable is in the expo...

TermDefinitionExample Exponential EquationAn equation in which the variable appears in the exponent.5^(2x) = 625 is an exponential equation. The variable is 'x'. LogarithmThe inverse operation of exponentiation. The expression log_b(y) asks the question: 'What exponent do I need to raise the base 'b' to in order to get 'y'?'log₂(8) = 3, because 2³ = 8. Common LogarithmA logarithm with base 10. It is typically written as log(x) without an explicit base.log(1000) = 3, because 10³ = 1000. Natural LogarithmA logarithm with the number 'e' (Euler's number, approximately 2.71828) as its base. It is written as ln(x).ln(e²) = 2. The natural log is the inverse of the exponential function e^x. Inverse Property of LogarithmsLogarithms and exponen...

Equality Property of Logarithms If M = N (and M, N > 0), then log_b(M) = log_b(N). This property allows us to take the logarithm of both sides of an equation without changing the equality. This is the first step in solving most exponential equations. Power Rule of Logarithms log_b(M^p) = p * log_b(M) This is the most critical rule for this topic. It allows you to take an exponent (p) from inside a logarithm and move it to the front as a coefficient, effectively freeing the variable from the exponent. Change of Base Formula log_b(M) = log(M) / log(b) OR ln(M) / ln(b) This formula is used to evaluate logarithms with bases other than 10 or 'e' using a standard calculator. It allows you to convert any log into common or natural logs.