1. Common Logarithms *Find common logarithms and exponential values of numbers *Solve equations using common logarithms *Solve real-world applications with common logarithmic functions Because of their frequent use in real-world and applied problems, logarithms with base 10 are referred to as common logarithms . Common logs written without an indicated base are assumed to be base 10. log 10 x = log x The standard calculator log button assumes log base 10.

2. The next step is to use this understanding to solve exponential equations: Solve for x: 1. 6 3x = 81 *Use the properties of logarithms to pull the variable out of the equation! 2. 5 4x = 73 3. 6 x - 2 = 4 x

3. Hmm… That last one took a bit of work Is there any other way to solve an equation? Solve for x: This time by graphing! 2 x-1 = 5 x - 2 You are determining the value of x that makes both equations true. Graph the two functions separately and look for an intersection point.

4. Solve: 2 x - 1 = 5 x - 2 That’s our point! x ≈ 2.76

5. Base e and Natural Logarithm (Natural Logarithms are better for your health)B *Find natural logarithms of numbers *Solve equations using natural logarithms *Solve real-world applications with natural logarithmic functions Because e is frequently used as an exponential base, log base e is defined as the natural logarithm log e x = ln x

6. Natural Log is the inverse of exponential base e = 2.718 Properties of logarithms hold the same for ln x Here’s the question then: Why is ln e = 1? Why is e ln x = x?

7. Because natural functions and natural logarithmic functions are inverses, these two functions can be used to “undo” each other