Exponentials & Logarithms

Residuals & Regression

• A residual is the difference between an actual value and the model value

• residual = actual value - predicted value

• If a model (regression) for a given set of data is appropriate, the residual plot should appear without a pattern

Exponential & Logarithmic Properties

• change of base theorem: logb a = logc a/logc b

Exponential & Logarithmic Equations Properties

Exponential Applications

• standard model form: y = ab^x

• half life model: A = P(1/2)^t/h

• financial model: A = P(1 + r/n)^nt

• continuous mode: A = Pe^rt

• y = 2 × 3^x

• y = 2 x (e^ln3^x)

• A = R((1 + r/n)^nt - 1)/(r/n)) is the future value of an annuity

• measures how much a deposit will be worth with continuous amounts of money deposited each month

• P = R((1 + r/n)^nt -1)/(r/n)(r + r/n)^nt)

• measures the present value of an annuity

Semilog Plots

• If a function f shows exponential growth or decay, plotting (x, log(f(x))) will linearize the data

• y = a x b^x

• a = the predicted y intercept

• b = predicted growth/decay

• y = log a + xlogb

• log a = predicted log of the y intercept

• log b = predicted increase/decrease in log (dependent variable) per unit of independent variable