Download presentation

Presentation is loading. Please wait.

1
**3.3 Logarithmic Functions and Their Graphs**

We learned that, if a function passes the horizontal line test, then the inverse of the function is also a function. So, an exponential function f(x) = bx, has an inverse that is a function. This inverse is the logarithmic function with base b, denoted or

2
**Changing Between Logarithmic and Exponential Form**

If x > 0 and 0 < b ≠ 1, then if and only if This statement says that a logarithm is an exponent. Because logarithms are exponents, we can evaluate simple logarithmic expressions using our understanding of exponents.

3
**Evaluating Logarithms**

a.) because . b.) because . c.) because d.) because e.) because

4
**Basic Properties of Logarithms**

For 0 < b ≠ 1 , x > 0, and any real number y, logb1 = 0 because b0 = 1. logbb = 1 because b1 = b. logbby = y because by = by. because

5
**Evaluating Logarithmic and Exponential Expressions**

(a) (b) (c)

6
**Common Logarithms – Base 10**

Logarithms with base 10 are called common logarithms. Often drop the subscript of 10 for the base when using common logarithms. The common logarithmic function: y = log x if and only if 10y = x.

7
**Basic Properties of Common Logarithms**

Let x and y be real numbers with x > 0. log 1 = 0 because 100 = 1. log 10 = 1 because 101 = 10. log 10y = y because 10y = 10y . because log x = log x. Using the definition of common logarithm or these basic properties, we can evaluate expressions involving a base of 10.

8
**Evaluating Logarithmic and Exponential Expressions – Base 10**

(a) (b) (c) (d)

9
**Evaluating Common Logarithms with a Calculator**

See example 4 p. 303

10
**Solving Simple Logarithmic Equations**

Solve each equation by changing it to exponential form: a.) log x = 3 b.) a.) Changing to exponential form, x = 10³ = 1000. b.) Changing to exponential form, x = 25 = 32.

11
**Natural Logarithms – Base e**

Because of their special calculus properties, logarithms with the natural base e are used in many situations. Logarithms with base e are natural logarithms. We use the abbreviation “ln” (without a subscript) to denote a natural logarithm.

12
**Basic Properties of Natural Logarithms**

Let x and y be real numbers with x > 0. ln 1 = 0 because e0 = 1. ln e = 1 because e1 = e. ln ey = y because ey = ey. eln x = x because ln x = ln x.

13
**Evaluating Logarithmic and Exponential Expressions – Base e**

(a) (b) (c)

14
**Transforming Logarithmic Graphs**

Describe how to transform the graph of y = ln x or y = log x into the graph of the given function. a.) g(x) = ln (x + 2) The graph is obtained by translating the graph of y = ln (x) two units to the LEFT.

15
**Transforming Logarithmic Graphs**

Describe how to transform the graph of y = ln x or y = log x into the graph of the given function. b.) h(x) = ln (3 - x) The graph is obtained by applying, in order, a reflection across the y-axis followed by a transformation three units to the RIGHT.

16
**Transforming Logarithmic Graphs**

Describe how to transform the graph of y = ln x or y = log x into the graph of the given function. c.) g(x) = 3 log x The graph is obtained by vertically stretching the graph of f(x) = log x by a factor of 3.

17
**Transforming Logarithmic Graphs**

Describe how to transform the graph of y = ln x or y = log x into the graph of the given function. d.) h(x) = 1+ log x The graph is obtained by a translation 1 unit up.

18
**Measuring Sound Using Decibels**

The level of sound intensity in decibels (dB) is where (beta) is the number of decibels, I is the sound intensity in W/m², and I0 = 10 – 12 W/m² is the threshold of human hearing (the quietest audible sound intensity).

19
More Practice!!!!! Homework – Textbook p. 308 #2 – 52 even.

Similar presentations

OK

Exponents – Logarithms xy -31/8 -2¼ ½ 01 12 24 38 xy 1/8-3 ¼-2 ½ 10 21 42 83 The function on the right is the inverse of the function on the left.

Exponents – Logarithms xy -31/8 -2¼ ½ 01 12 24 38 xy 1/8-3 ¼-2 ½ 10 21 42 83 The function on the right is the inverse of the function on the left.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on second law of thermodynamics entropy Ppt on acid-base titration indicator Ppt on environmental awareness Ppt on any one mathematician fibonacci Ppt on review of related literature example Ppt on employment and unemployment in india Ppt on bioreactors for wastewater treatment Ppt on modern art by paul Ppt on viruses and bacteria pictures Ppt on non renewable energy resources