1. The Natural Logarithmic Function
Section 5.1
The Natural Logarithmic Function

2. THE NATURAL LOGARITHMIC FUNCTION
Definition: The natural logarithmic function is the function defined by
Remember this from the
graphing activity

3. THE DERIVATIVE OF THE NATURAL LOGARITHMIC FUNCTION
From the Fundamental Theorem of Calculus, Part 1, we see that
Remember we discussed
this in class

4. LAWS OF LOGARITHMS Remember these rules for logarithms.
If x and y are positive numbers and r is a rational number, then

5. PROPERTIES OF THE NATURAL LOGARITHMIC FUNCTION
Using calculus, we can describe
the natural logarithmic function.
Remember x>0
1. ln x is an increasing function, since
2. The graph of ln x is concave downwards, since

6. THEOREM
This is consistent with what
we know about the graph of ln(x)

7. THE DERIVATIVE OF THE NATURAL LOGARITHM AND THE CHAIN RULE
We introduced this in class.

8. ANTIDERIVATIVES INVOLVING THE NATURAL LOGARITHM
Theorem:
Remember the domain of the natural log is positive real numbers.

9. ANTIDERIVATIVES OF SOME TRIGONOMETRIC FUNCTIONS
Memorize these

10. LOGARITHMIC DIFFERENTIATION
How can we use this information to help us solve problems?
Take logarithms of both sides of an equation y = f (x) and use the laws of logarithms to simplify.
Differentiate implicitly with respect to x.
Solve the resulting equation for y′.

11. Example: Differentiate y=ln(3x2-2)3
Rewrite: y=3ln(3x2-2)
y’ = 3
ln(3x2-2)

12. Example: Differentiate y=ln(3x2-2)3
Rewite: y=3ln(3x2-2)
y’ = 3
ln(3x2-2)