Solve exponential equations using natural logarithms

Learning Objectives Identify exponential equations that require logarithms for their solution. Apply the natural logarithm to both sides of an exponential equation. Use the Power Rule of Logarithms to bring an exponent down as a coefficient. Solve for a variable exponent by isolating it algebraically. Calculate the approximate decimal value of a logarithmic expression using a calculator. Verify their solution by substituting it back into the original equation. How long would it take for a bacterial culture to double, or for an investment to triple? 📈 To find the 'time' in these growth problems, you need to solve for a variable in the exponent! This tutorial will teach you a powerful technique to solve for variables trapped in exponents. You will learn how to use...

TermDefinitionExample Exponential EquationAn equation where the variable you need to solve for is located in the exponent.In the equation 5^(x) = 25, 'x' is the variable in the exponent. Natural Logarithm (ln)A logarithm with a special base called 'e'. The natural logarithm, written as ln(x), asks the question: 'e to what power equals x?'.ln(7.389) ≈ 2, because e^2 ≈ 7.389. Base 'e'An irrational and transcendental number, approximately equal to 2.71828. It is often called Euler's number and is the base of the natural logarithm. It appears naturally in many models of continuous growth.The function f(x) = e^x represents continuous exponential growth. Inverse Property of Natural LogarithmsThe natural logarithm and the exponential function with bas...

One-to-One Property of Logarithms If ln(M) = ln(N), then M = N. This property allows us to take the natural logarithm of both sides of an equation, preserving the equality. This is the foundational step for solving exponential equations. Power Rule of Logarithms ln(M^p) = p * ln(M) This is the most critical rule for this topic. It allows you to take a variable exponent 'p' and move it outside the logarithm, turning an exponential problem into a linear one. Inverse Property ln(e^x) = x A special case used when the exponential equation has a base of 'e'. Applying the natural log directly cancels out the 'e', immediately freeing the exponent.