1. AP Calculus AB Chapter 5, Section 1 ish
Natural Logarithmic Functions: Differentiation

2. The Natural Logarithmic Function
Evaluate 𝑥

3. The Natural Logarithmic Function
Definition of the Natural Logarithmic Function:
ln 𝑥 = 1 𝑥 1 𝑡 𝑑𝑡 , 𝑥>0
The domain of the natural logarithmic function is the set of all positive real numbers.

4. Theorem: Properties of the Natural Logarithmic Function
The natural logarithmic function has the following properties:
The domain is (0, ∞) and the range is (-∞, ∞).
The function is continuous, increasing, and one-to-one.
The graph is concave downward.

5. Theorem: Logarithmic Properties
If a and b are positive numbers and n is rational, then the following properties are true:
ln 1 =0
ln (𝑎𝑏) = ln 𝑎 + ln 𝑏
ln 𝑎 𝑛 =𝑛 ln 𝑎
ln 𝑎 𝑏 = ln 𝑎 − ln 𝑏

6. Expanding Logarithmic Expressions
ln 𝑥 𝑥 2 +1

7. Condensing Logarithmic Expressions
2 ln 𝑥 ln (𝑥−2)

8. The Number e The number e is the base of ln.
e and ln are inverses of each other.
In the equation ln 𝑥 =1, the value of x to make this statement true is e.
e is irrational and has the decimal approximation 𝑒≈

9. Definition of e
The letter e denotes the positive real number such that
ln 𝑒 = 1 𝑒 1 𝑡 𝑑𝑡 =1

10. The Derivative of the Natural Logarithmic Function
Let u be a differential function of x
𝑑 𝑑𝑥 ln 𝑥 = 1 𝑥 , 𝑥>0
𝑑 𝑑𝑥 ln 𝑢 = 1 𝑢 𝑑𝑢 𝑑𝑥 = 𝑢′ 𝑢 , 𝑢>0

11. Derivative Involving Absolute Value
If u is a differentiable function of x such that 𝑢≠0, then
𝑑 𝑑𝑥 ln 𝑢 = 𝑢′ 𝑢

12. Differentiation of Logarithmic Functions
𝑦= ln ln 𝑥 , 𝑓𝑖𝑛𝑑 𝑦′

13. Differentiation of Logarithmic Functions
𝑑 𝑑𝑥 ln 𝑥 3

14. Differentiation of Logarithmic Functions
𝑦= ln cos 𝑥 , 𝑦 ′ =

15. Logarithmic Properties as Aids to Differentiation
Differentiate: 𝑓 𝑥 = ln 𝑥 𝑥 𝑥 3 −1

16. Show that 𝑦=𝑥 ln 𝑥 −4𝑥 is a solution to the differential equation 𝑥+𝑦−𝑥 𝑦 ′ =0

17. Finding Relative Extrema
Locate the relative extrema of
𝑦= ln ( 𝑥 2 +2𝑥+3)

18. Ch 5.1 Homework
Pg 329 – 330, #’s: 7 – 10, 15, 21, 27, 29, 33, 41, 49, 55, 61, 71, 75, 79