LogarithmsSection 3.3a and b!!!

Homework: p. 308 1-35 odd, 41-57 odd

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 Logarithmicand Exponential FormIf 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!!!

Basic 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 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…

Common 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 ofCommon 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 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 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 LogarithmicEquations

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???

Natural LogarithmsNatural Logarithms

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

Basic Properties of Natural Logarithms

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

Cool Practice Problems

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…

Cool Practice Problems

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

Cool Practice Problems

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

Analysis of the Natural Logarithmic Function:Analysis of the Natural Logarithmic Function:

The graph: lnf x x

Domain: 0, Range: , Continuous on 0,Increasing on 0,

No Symmetry Unbounded

No Local Extrema No Horizontal Asymptotes

Vertical Asymptote: 0x

End Behavior: lim lnx

x

Analysis of the Natural Logarithmic FunctionAnalysis of the Natural Logarithmic Function

The graph: lnf x xNote: Any other logarithmicNote: Any other logarithmicfunctionfunction logbg x xwith with bb > 1 has the same domain, > 1 has the same domain,range, continuity, inc. behavior,range, continuity, inc. behavior,lack of symmetry, and otherlack of symmetry, and othergeneral behavior of the naturalgeneral behavior of the naturallogarithmic function!!!logarithmic function!!!

Describe how to transform the graph of y = ln(x) or y = log(x)into the graph of the given function. Sketch the graph by handand support your answer with a grapher.

1. ln 2g x x Trans. left 2 Trans. left 2 The graph? The graph?

2. ln 3h x x Reflect across the Reflect across the yy-axis,-axis,Trans. right 3 Trans. right 3 The graph? The graph?

ln 3x

Describe how to transform the graph of y = ln(x) or y = log(x)into the graph of the given function. Sketch the graph by handand support your answer with a grapher.

3. 3logg x xVert. stretch by 3 Vert. stretch by 3 The graph? The graph?

4. 1 logh x x Trans. up 1 Trans. up 1 The graph? The graph?

Describe how to transform the graph of y = ln(x) or y = log(x)into the graph of the given function. Sketch the graph by handand support your answer with a grapher.

5. 2ln 2 2 3g x x

Trans. right 1, Horizon. shrink by 1/2,Trans. right 1, Horizon. shrink by 1/2,Reflect across both axes, Vert. stretch by 2,Reflect across both axes, Vert. stretch by 2,Trans. up 3 Trans. up 3 The graph??? The graph???

2ln 2 1 3x

Graph the given function, then analyze it for domain, range,continuity, increasing or decreasing behavior, boundedness,extrema, symmetry, asymptotes, and end behavior.

1. log 2f x x D: 2, R: ,

Continuous Dec: 2, No Symmetry Unbounded No Local Extrema

Asy: 2x E.B.: limx

f x

Graph the given function, then analyze it for domain, range,continuity, increasing or decreasing behavior, boundedness,extrema, symmetry, asymptotes, and end behavior.

2. 5ln 2 3f x x

D: , 2 R: ,

Continuous Dec: , 2 No Symmetry Unbounded No Local Extrema

Asy: 2x E.B.: limx

f x