Convert between natural exponential and logarithmic form

Learning Objectives Define the natural logarithm (ln) and its relationship to the base e. Identify the base, exponent, and argument in both natural exponential and logarithmic expressions. Accurately convert an equation from natural exponential form (y = e^x) to its equivalent natural logarithmic form (x = ln(y)). Accurately convert an equation from natural logarithmic form (x = ln(y)) to its equivalent natural exponential form (y = e^x). Solve simple equations for a variable by converting between the two forms. Evaluate basic natural logarithmic expressions by converting them to exponential form. Ever wonder how scientists model things that grow constantly, like a bacterial culture or money in a special savings account? 🌿💰 It all comes down to a special number and its inv...

TermDefinitionExample The Number e (Euler's Number)An irrational mathematical constant, approximately equal to 2.71828. It is the base of the natural logarithm and is used to model continuous growth and decay.The value of e^1 is approximately 2.71828. Natural Exponential FormAn equation written with a base of 'e' raised to an exponent. The general form is y = e^x.The equation P = 100e^(0.05t) is in natural exponential form. Natural Logarithm (ln)A logarithm with a base of 'e'. The expression ln(x) is a shorthand way of writing logₑ(x). It answers the question: 'To what power must e be raised to get x?'ln(e^2) = 2, because the power you must raise e to in order to get e^2 is 2. Natural Logarithmic FormAn equation written using the natural logarithm. The g...

Conversion: Exponential to Logarithmic If e^x = y, then ln(y) = x Use this rule to rewrite an equation from exponential form into logarithmic form. This is essential when you need to solve for a variable that is in the exponent. Conversion: Logarithmic to Exponential If ln(y) = x, then e^x = y Use this rule to rewrite an equation from logarithmic form into exponential form. This is the key step to solving for a variable that is inside a natural logarithm.