Logarithmic Functions Section 3.2 Logarithmic Functions

Logarithmic Function The inverse of f(x) = bx Denoted logb x “log base b of x”

Example 1 A. Evaluate log216. B. Evaluate C. Evaluate log1717

Example 2 A. Evaluate log8 512. B. Evaluate 22log22 15.2

Example 3 A. Evaluate log 10,000 B. Evaluate 10log 12. C. Evaluate log 14. D. Evaluate log (–11).

Natural Logarithm A logarithm with base e or loge Denoted ln (“natural log”) y = lnx is the inverse of y = ex The properties of logarithms hold true for natural logarithms.

Example 4 A. Evaluate ln e4.6. B. Evaluate ln (–1.2). C. Evaluate eln 4.

Graphing Logarithmic Functions

Example 5 A. Sketch and analyze the graph of f (x) = log2 x. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing.

Example 5 B. Sketch and analyze the graph of Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing.

Transformation Equation Description Vertical translation   g(x) = + k g(x) = – k *Shifts the graph of f(x) = upward k units. *Shifts the graph of f(x) = downward k units. Horizontal g(x) = *Shifts the graph of f(x) = to the left h units. Vertical asymptote x = h *Shifts the graph of f(x) = to the right h units. Reflection g(x) = – *Reflects the graph of f(x) = about the x-axis about the y-axis

Example 6 A. Use the graph of f (x) = log x to describe the transformation that results in p (x) = log (x + 1). Then sketch the graph of the function.

Example 6 B. Use the graph of f (x) = log x to describe the transformation that results in m (x) = –log x – 2. Then sketch the graph of the function.

Example 7 A. EARTHQUAKES The Richter scale measures the intensity R of an earthquake. The Richter scale uses the formula R where a is the amplitude (in microns) of the vertical ground motion, T is the period of the seismic wave in seconds, and B is a factor that accounts for the weakening of seismic waves. A city is not concerned about earthquakes with an intensity of less than 3.5. An earthquake occurs with an amplitude of 125 microns, a period of 0.33 seconds, and B = 1.2. What is the intensity of the earthquake? Should this earthquake be a concern for the city?

Example 7 B. EARTHQUAKES The Richter scale measures the intensity R of an earthquake. The Richter scale uses the formula R where a is the amplitude (in microns) of the vertical ground motion, T is the period of the seismic wave in seconds, and B is a factor that accounts for the weakening of seismic waves. Earthquakes with an intensity of 6.1 or greater can cause considerable damage to those living within 100 km of the earthquake’s center. Determine the amplitude of an earthquake whose intensity is 6.1 with a period of 3.5 seconds and B = 3.7.

Example 8 Find the inverse of the equation:

Homework for section 3.2 p. 178; 1-15 odd, 28-30, 34-35, 43, 47-49 H