1. Unit 6 Lesson 1
Natural Logs

2. y = loga x if and only if x = a y.
For x  0 and 0  a  1,
y = loga x if and only if x = a y.
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 = a y
A logarithm is an exponent!
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. In Calculus, we work almost exclusively with natural logarithms!5 –5
y = ln x
The function defined by f(x) = loge x = ln x
(x  0, e )
is called the natural logarithm function.
y = ln x is equivalent to e y = x
In Calculus, we work almost exclusively with natural logarithms!

5. Examples

6. Examples

7. Example: Solution: 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. Integrating is going backwards
Finding the anti-derivative using natural logs is fun, fun, fun 

23. Integrals of 6 basic trig functions