Section 3.3a!!!. First, remind me… What does the horizontal line test tell us??? More...

Section 3.3a!!!

First, remind me…

What does the horizontal line test tell us???

More specifically, what does it tell us about the function

xf x bThis function has an inverse

that is also a function!!!

This inverse is called thelogarithmic function with base b.

Notation:

1 logbf x x

Changing Between LogarithmicChanging Between Logarithmicand Exponential Formand Exponential Form

If x > 0 and 0 < b = 1, then

logby x if and only ifyb x

Important Note: The “linking statement” says thata logarithm is an exponent!!!a logarithm is an exponent!!!

Basic Properties of LogarithmsBasic Properties of Logarithms

For 0 < b = 1, x > 0, and any real number y,

log 1 0b 1.0 1b because

log 1b b2.1b bbecause

log yb b y3.

y yb bbecause

logb xb x4. log logb bx xbecause

Evaluating LogarithmsEvaluating LogarithmsEvaluate each of the following.

2log 81. 3

3log 32. 1 2

7log 73. 1

9log 14. 06log 1165. 11

5

1log

256. 2

What’s true about the (x, y) pairs and graphs of inverse functions?

–3

x 2xf x x 12logf x x

–2

–1

0

1

2

3

–3

–2

–1

0

1

2

3

1/8

1/4

1/2

1

2

4

8

1/8

1/4

1/2

1

2

4

8

Now, let’s plot these points and discuss the graphs…Now, let’s plot these points and discuss the graphs…

Common LogarithmsCommon Logarithms

Common Logarithm – logarithm with a base of 10

(very commonly used because of our base 10 number system!)

For common logarithms, we can drop the subscript:

logy x if and only if 10y x

Basic Properties ofBasic Properties ofCommon LogarithmsCommon Logarithms

Let x and y be real numbers with x > 0.

log1 01.010 1because

log10 12.110 10because

log10 y y3. 10 10y ybecause

log10 x x4. log logx xbecause

More Evaluating LogarithmsMore Evaluating LogarithmsEvaluate each of the following.

log1001. 2 0.367log102. 0.367

1log1000

3. 3 5log 1004.2

5

Note: The LOG key on your calculator refers to the common logarithm…

Using Your CalculatorUsing Your CalculatorUse a calculator to evaluate the logarithmic expression if it isdefined, and check your result by evaluating the correspondingexponential expression.

1. log34.5 1.537... b/c1.537...10 34.5

2. log0.43 0.366... b/c0.366...10 0.43

3. log 3 is undefined can you explain why ?

Solving Simple LogarithmicSolving Simple LogarithmicEquationsEquations

Solve the given equations by changing to exponential form.

1. log 3x Exp. Form:

310x 1000x

2. 2log 5x Exp. Form:

52x 32x

1lim 1

x

xe

x

What is the definition of the natural base???

log lne x x

Natural Logarithm – a logarithm with base e

Notation: ln

That is,

Back to our inverse relationship:

lny x if and only ifye x

ln1 0Let x and y be real numbers with x > 0.

1. because0 1e

ln 1e2. because1e e

ln ye y3. becausey ye e

ln xe x4. because ln lnx x

ln e

Evaluate each of the following without a calculator.

1. 1 25ln e2. 5

ln 4e3. 4

Note: The LN key on your calculator refers to the natural logarithm…

ln 23.5

Use a calculator to evaluate the given logarithmic expressions,if they are defined, and check your result by evaluating thecorresponding exponential expression.

1. 3.157... because3.157... 23.5e

ln 5.432. is undefined!!! Why???

ln 0.483. 0.733... because0.733... 0.48e

log 5x

Solve each of the given equations by changing them toexponential form.

1.

xx = 100,000 = 100,000

log 2x 3.

xx = = 0.01 = = 0.01

ln 1x 2.

xx = = 0.368… = = 0.368…

11

100100

11

ee

ln 2.5x 4.

xx = = ee = 12.182… = 12.182…2.52.5