key concepts

3.1 - evaluate and graph exponential functions

• the exponential function f with base a is denoted by f(x)=a^x, where a > 0, a =/= 1 and x is any real number

• the graphs of the exponential functions f(x) = a^x and f(x) = a^-x have one y-intercept, one horizontal asymptote (the x-axis), and are continuous

• the natural exponential function is f(x) = e^x, where e is the constant 2.718281828… its graph has the same basic characteristics as the graph of f(x) = a^x

3.2 - evaluate and graph logarithmic functions

• for x > 0, a > 0, and a=/=1, y = log base a of x if and only if x = a^y. the function given by f(x) = log base a of x is called the logarithmic function with base a.

• the graph of the logarithmic function f(x) = log base a of x, where a > 1, is the inverse of the graph of f(x) = a^x, has one x-intercept, one vertical asymptote (the y-axis), and is continuous.

• for x > 0, y = ln x if and only if x = e^y. the function given by f(x) - log base e of x = ln x is called the logarithmic function. its graph has the same basic characteristics as the graph of f(x) = log base a of x.

3.2 properties of logarithms

• log base a of 1 = 0 and ln 1 = 0

• log base a of a = 1 and ln e = 1

• log base a of a^x = x, a^(log base a of x) = x; ln e^x = x, e^(ln x) = x

• if log base a of x = log base a of y, then x = y. If ln x = ln y, then x = y.

3.3 change-of-base formulas and properties of logarithms

• let a, b and x be positive real numbers such that a =/= 1 and b =/= 1. Then log base a of x can be converted to a different base using the formula log base a of x = (log base b of x) / (log base b of a)

• let a be a positive number such that a=/=1, and let n be a real number. if u and v are positive real numbers, the following properties are true:

  • product property

    • log base a of (uv) = log base a of u + log base a of v

    • ln(uv) = ln u + ln v

  • quotient property

    • log base a of (u/v) = log base a of u - log base a of v

    • ln(uv) = ln u - ln v

  • power property

    • log base a of u^n = n log base a of u

    • ln u^n = n ln u

3.4 solve exponential and logarithmic equations

• rewrite the original equation to allow the use of the one-to-one properties 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

3.5 use nonalgebraic models to solve real-life problems

• exponential growth model: y = ae^bx, b>0

• exponential decay model: y = ae^-bx, b>0

• gaussian model: y = ae^(-(x-b)^2/c)

• logistic growth model: y = a / (1+be^-rx)

• logarithmic models: y = a + b ln x, y = a + b log x

3.6 fit nonlinear models to data

• create a scatter plot of the data to determine the type of model (quadratic, exponential, logarithmic, power, or logistic) that would best fit the data

• use a calculator or computer to find model

• the model whose y-values are closest to the actual y-values is the one that fits best