logarithmic functions. examples properties examples

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.