1. Logarithmic, Exponential, and Other Transcendental Functions
Chapter 5
Logarithmic, Exponential, and Other Transcendental Functions

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!

4. Definition of the Natural Logarithmic Function

5. Theorem 5.1 Properties of the Natural Logarithmic Function

6. Natural Log

7. Natural Log Passes through (1,0) and (e,1).
You can’t take the log of zero or a negative.
(Same graph 1 unit right)

8. Theorem 5.2 Logarithmic Properties

9. Properties of Natural Log:
Expand:
Write as a single log:

10. Properties of Natural Log:
Expand:
Write as a single log:

11. Definition of e

12. Theorem 5.3 Derivative of the Natural Logarithmic Function

13. 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:

14. Example

15. Example

16. Example
Product Rule

17. Example

18. Example

19. Example

20. Example

21. Theorem:

22. Theorem:

25. Theorem 5.4 Derivative Involving Absolute Value

27. Try Logarithmic Differentiation.

30. 4. Show that is a solution to the statement .

31. 4. Show that is a solution to the statement .

32. At (1, 3) the slope of the tangent is 2
Find the equation of the line tangent to: at (1, 3)
At (1, 3) the slope of the tangent is 2

33. Find the equation of the tangent line to the graph of the function
at the point (1, 6).