Logarithmic Functions Section 11.3 Logarithmic Functions

Definition of a Logarithm For b > 0, b ≠ 1, and a > 0, the logarithm logb (a) is the number k such that bk = a. We call b the base of the logarithm. Rewrite each logarithmic expression as an exponential and simplify. Example

Definition of a Logarithm Finding Common Logarithm Solution

Definition of a Logarithm Common Logarithm Definition A common logarithm is a logarithm with base 10. We write log(a) to represent log10(a). Rewrite each logarithmic expression as an exponential and simplify. Example

Definition of a Logarithm Common Logarithm Solution

Properties of Logarithms Properties and Definitions of Logarithms Properties For b > 0 and b ≠ 1, logb (b) = 1 logb (1) = 0 A logarithmic function, base b, is a function that can be put into the form where b> 0 and b ≠ 1. Definitions

Properties of Logarithms For an exponential function For a logarithmic function In words, are inverse functions.

Find the inverse of the function. 2. Properties of Logarithms Finding an Inverse Function Example Find the inverse of the function. 2. Solution

Properties of Logarithms Evaluating f and f-1 Example Let Solution

Graphing a Logarithmic Function Example Sketch the graph of Applying the four-step method: Step 1: Sketch a curve of f: Step 2: Choose several points on the graph of f: (-1, 1/3), (0, 1), (1, 3) and (2, 9). Solution

Graphing a Logarithmic Function Solution Example Step 3: For each point (a, b) chosen in step 2, plot point (b, a): We plot (1/3, -1), (1, 0), (3, 1) and (9, 2). Step 4: Sketch the curve containing the points in Step 3.

Using Logarithms to Model Authentic Situations Logarithm Uses Logarithm scales are used for: Amplitudes or earthquakes Noise levels of sounds pH values Earthquake Richter scale, R, is given by where A is the amplitude and A0 is the reference amp. Definition

Using Logarithms to Model Authentic Situations Richter Numbers Example In 1906, an earthquake in San Francisco had an amplitude times the reference amplitude A0. In 1989, an earthquake had an amplitude times A0. Find the Richter number of both earthquakes. Find the ratio of the amplitudes of the 1906 and 1989 earthquakes.

1. The Richter number of the 1906 earthquake is Using Logarithms to Model Authentic Situations Richter Numbers Solution 1. The Richter number of the 1906 earthquake is 2. The Richter number of the 1989 earthquake is

Using Logarithms to Model Authentic Situations Richter Numbers Solution Solution Continued 3. The ratio of the amplitudes is: So, the 1906 earthquake had an amplitude 25 times greater than that of the 1989 earthquake.