Logarithmic Functions
is always increasing or always decreasing Exponential functions are 1-1 neither x or y repeats1-1 functions have inverses logarithmic function with base LOGARITHMS ARE EXPONENTS!When you find a logarithm, you are finding an exponent of the base.
(y is the exponent of that yields x)
Examples
1.(Think: )
2.
Graph and together
4
Properties
Recall exponent properties
Log Properties
,
Examples
1.
2.
Homework: p. 365 #1-5, 11-13, 15, 16
4
Natural Log Function
*Same properties apply as regular log functions.Graph and
1
Solve for x
To undo logarithms, do the inverse.
Homework: p. 365 #6, 7, 14, 17, 18, 20, 37-48
Change of Base Formula
Every log function can be written as a quotient of two natural logs.
𝐵𝑦=𝑥
Derivatives of Natural Log Functions
In general,
2 𝑥 ln 𝑥+¿
Homework: p. 374 #1-4, 7, 8
Derivatives of Any Log Functions
Homework: p. 374 #5, 9, 11-22
Integrals with Logs
ln ¿ 𝑥∨¿+𝑐 ¿
Homework: p. 374 #67-78
Test Review
1. Find Find derivatives.2. 3. 4. 5. 6. 7.
Find Antiderivatives and Definite Integrals8. 9. 10. 11. 12.
Review Answers
1. 2. 3. 4. 5. 6. 7.
8. 9. 10. 11. 12.
Rules for Test
Derivative rules Product Quotient Chain
Derivatives of trig functions Properties of logarithms
Integration Rules U-substitution
Integrals of trig functions
10 questions – 8 multiple choice
More Practice
Find derivatives
1.
2.
Find integrals
5.
Logarithmic Differentiation
Can be used to simplify very complicated derivatives.Examples: Product and quotient rule together Variable raised to a variable
Steps:1. Take the natural log of both sides.2. Differentiate both sides implicitly with respect to x.3. Solve the resulting equation for .
Example
Find the derivative of .1. Ln of both sides
2. Differentiate both sides
3. Solve for
Example
Find the derivative of
On Your Own
1. 2. 3. 4.
Homework: p. 375 #33-44*Use Ex. 2 for #43*Use #34 for #44
p. 375 Evens
34.
36.
38.
40.
42.
44.