Logarithmic Functions

How Tall Are You?

Objective I can identify logarithmic functions from an equation or graph. I can graph logarithmic functions using a graphing utility and identify asymptotes, intercepts, domain and range.

Warmup Find the inverse of f(x) = x 2 Find the inverse of f(x) = 2 x

Logarithmic Function An logarithmic function is a function of the form f(x) = log b (x), where b is > 0, is a positive constant (called the base) and not equal to 1, and x >0. y = log b (x) is equivalent to b y =x y = log b (x) Logarithmic form b y =xExponential form Logarithmic functions are either strictly increasing or strictly decreasing. Logarithmic functions have asymptotes.

How Tall Are You? Percent of adult height obtained by a male x years old can be modeled by f(x) = log 10 (x+1) What percent of adult height is an eight year old male? What percent of adult height is a 17 year old male?

Logarithmic Function The inverse function of the exponential function with base b is called the logarithmic function with base b y x

Logarithmic Function Example: Draw and label f(x) = 4 x and g(x) = log 4 (x) Identify domain, range and asymptotes y x

Logarithmic Function Example: Draw and label f(x) = (1/2) x and g(x) = log (1/2) (x) Identify domain, range and asymptotes y x

Characteristics of Logarithmic Functions For logarithmic functions of the form f(x) = log b (x) Domain: {x | x > 0} all real numbers (0, ∞) Range: {y | -∞ ≤ y ≤ ∞} all real numbers (-∞, ∞) Graph of f(x) = log b (x) passes thru (1,0) [ f(1) = log b (1) = 0] No Y-intercept X-intercept is (1,0) If b > 1, f(x) = log b (x) goes up to right, increasing function. If 0 < b < 1, f(x) = log b (x) goes down to right, decreasing function. f(x) = log b (x) is one to one and has an inverse function f(x) = log b (x) is the inverse function of f(x) = b x – If f(x) = b x, then f -1 (x) = log b (x) Graph of f(x) = log b (x) approaches but does not touch y axis. Vertical asymptote : x=0 (y axis)

Transformations of Logarithmic Functions TransformationEquationDescription Vertical Translation Horizontal Translation Reflection Vertical Shrinking or Stretching Horizontal Shrinking or Stretching

Transformation Examples y x y x