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.
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
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.
Solve: 2 x - 1 = 5 x - 2 That’s our point! x ≈ 2.76
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
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?
Because natural functions and natural logarithmic functions are inverses, these two functions can be used to “undo” each other
Write equivalent Expressions <ul><li>Write a equivalent exponential or logarithmic equation. </li></ul><ul><li> </li></ul><ul><li> </li></ul>
Using e and ln to solve problems and equations: <ul><li>Solve for x: </li></ul><ul><li>3 2x = 7 x – 1 2. 6.5 = -16.25 ln x </li></ul><ul><li>3. 18 = e 3x 5. 10 = 5 e 5x </li></ul>