Skowhegan Area High School
 Topics Introduction Key Terms Lesson 1: Reasoning with Exponential and Logarithmic Functions Lesson 2: Reasoning with Trigonometric functions Solving Trigonometric Equations Conclusion
 FUNCTIONS & SYMBOLIC REASONING Reasoning with Exponential and Logarithmic Functions Introduction: In a nutshell, the concept behind this lesson is to review and extend important ideas involving symbolic reasoning with exponential expressions and equations, as well as to learn about natural logarithms and how to solve logarithmic equations. Objectives: -To understand the number “e” and natural logarithms. -To solve logarithmic equations involving either common logarithms or natural logarithms.   Log Functions with Different Bases   So what is the Natural log? The man who came up with the number 2.71828182846 was named John Napier, but the man who gave that number the symbol “e” was named Leonhard Euler. Euler is commonly referred to as the “inventor” of “e” and he gave it that symbol when he was trying to work on the problem of continually compounded interest. Other uses for“e” Besides being used for problems involved in continually compounded interest, “e”can also be used to show bacterial growth. If the problem has the bacteria increasing by 150% during a given unit of time, values will ultimately approach an asymptote involving "e". The value "e" is also used for modeling simple population growth, such as rabbit offspring growth, or half-life showing exponential decay. So now that “e” is known to you, let’s move on to solving logarithmic equations. Let’s try another one... All right, one more....   And now here’s an example of solving a natural log...   So, as you can see, solving natural logarithms is really no different than solving any other logarithms. Here’s one more example of a natural log This really doesn’t do much for you, however, because you can press the calculator key, ln4, and get the same answer. Use natural logs to solve the next equation Now use common logs to solve the same problem What do you notice?