Natural Logs

1. Unit 6 Natural Logs

2. A logarithm is an exponent! For x 0 and 0  a  1, y = loga x if and only if x = ay. The function given by f(x) = loga x is called the logarithmic function with base a. Every logarithmic equation has an equivalent exponential form: y= loga x is equivalent to x =ay A logarithmic function is the inverse function of an exponential function. Exponential function: y = ax Logarithmic function: y = logax is equivalent to x = ay

3. y (x  0, e 2.718281) x 5 –5 y = ln x is equivalent to ey = x y = ln x The function defined by f(x) = logex = ln x is called the natural logarithm function. In Calculus, we work almost exclusively with natural logarithms!

5. Examples

6. Examples

7. Derivative of Logarithmic Functions The derivative is Notice that the derivative of expressions such as ln|f(x)| has no logarithm in the answer. Example: Solution:

9. Example

10. Example

11. Example Product Rule

12. Example

13. Example

14. Example

15. Example

16. Finding the anti-derivative using natural logs is fun, fun, fun  Integrating is going backwards

23. Integrals of 6 basic trig functions