Precalculus Exponential & Logarithmic Functions
Exponential Functions • The exponential function f with base a is defined as f(x) = ax . where a > 0, a ≠ 1, and x is any real number. • For a > 1, the graph of y = ax is ___________ over its domain. • For a > 1, the graph of y = a-x is __________ over its domain. • For the graph of y = ax or y = a-x, a > 1, the domain is _______, the range is ________, and the intercept is __________. Also, both graphs have __________ as a horizontal asymptote.
Natural Exponential Functions • The natural exponential function f has base e and is defined as f(x) = ex . • The graph of y = ex is ___________ over its domain. • The graph of y = e-x is __________ over its domain. • For the graph of y = ex or y = e-x, the domain is _______, the range is ________, and the intercept is __________. Also, both graphs have __________ as a horizontal asymptote.
Compounding Interest • After t years, the balance A in an account with principal P and annual interest rate r (expressed as a decimal) … • For n compoundings per year: • For continuous compounding: • Note: Continuous Compounding yields more than quarterly or monthly compounding …
Examples • f(-x) = Reflect about the y-axis … • -f(x) = Reflect about the x-axis … • Given y = ax … • __________ is a reflection about the y-axis. • __________ is a reflection about the x-axis.
Examples • Graph the following functions …
Examples • Determine if the following functions are the same …
Examples • Radium 226 has a half life of 1620 years. Its mass (Q) is given by the following equation (t = years) … • Find the initial quantity. • Find the quantity after 1000 yrs. • Graph the function over t = 0 to t = 5000. • When will the quantity be 0 grams?
Logarithmic Functions • The logarithmic function with base a is defined as y = logax for x > 0, a > 0, a ≠ 1, if and only if x = ay. • The equation x = ay in exponential form is equivalent to the equation y = logax in logarithmic form. • The logarithmic function with base a is the inverse of the exponential function f(x) = ax. • Example: Use the definition of a logarithmic function to evaluate log5125 …
Properties of Logarithms • loga1 = ____________ (because a0 = 1) • logaa = ____________ (because a1 = a) • logaax = ____________ (Inverse Properties) • alogax = ____________ (Inverse Properties • If logax = logay , then _____________ (1-1 Property)
Graphs of Logarithms • To sketch y = logax , use the fact that the graphs of inverse functions are reflections of each other in the line y = x. • For the graph of y = logax , a > 1, the domain is _________, the range is __________, and the intercept is __________. • Also, the graph has __________ as a vertical asymptote.
Examples • Graph the following functions …
Examples • Graph the following functions …
Properties of Natural Logarithms • ln1 = ____________ (because e0 = 1) • lne = ____________ (because e1 = e) • lnex = ____________ (Inverse Properties) • elnx = ____________ (Inverse Properties • If lnx = lny , then _____________ (1-1 Property)
Examples • Write the logarithmic equation in exponential form … • Write the exponential equation in logarithmic form …
Examples • Solve the equation for x …
Examples • Find the domain, vertical asymptote, and x-intercept of the following functions …
Properties of Logarithms • Common Logarithms = base 10 • Natural Logarithms = base e
Change of Base Formula • Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. Then logax can be converted to a different base using any of the following formulas.
Examples • Evaluate the following logarithms using the change-of-base formula …
Examples • Rewrite the logarithm as a multiple of a common logarithm and a natural logarithm …
Properties of Logarithms • Let a be a positive number such that a ≠ 1, and let n be a real #. If u and v are positive real #’s, the following are true …
Examples • Use the properties of logarithms to write the expression as a sum, difference, and/or constant multiple of a logarithm …
Examples • Write the expression as the logarithm of a single quantity …
Examples • Find the exact value of the logarithm without using a calculator …
Solving Exponential & Logarithmic Equations • One-to-One Properties … ax = ay if and only if x = y logax = logay if and only if x = y • Inverse Properties …
Solving Exponential & Logarithmic Equations • Rewrite the equation in a form to use the one-to-one properties of exponential or logarithmic functions. • Rewrite an exponential equation in logarithmic form and apply the inverse property of logarithmic functions. • Rewrite a logarithmic equation in exponential form and apply the inverse property of exponential functions.
Examples • Solve the following exponential equations …
Exponential & Logarithmic Models • Exponential Growth • Exponential Decay • Gaussian Model • Logistic Growth • Logarithmic Model
