11

Algebra 2: Section 8.4Algebra 2: Section 8.4

Logarithmic FunctionsLogarithmic Functions(Day 1)(Day 1)

Solving for “x”Solving for “x”

AdditionAddition x – 3 = 5x – 3 = 5 SubtractionSubtraction 3 + x = 93 + x = 9 MultiplicationMultiplication 1/2x = 41/2x = 4 DivisionDivision 5x = 255x = 25 PowerPower xx33 = 27 = 27 RootsRoots If “x” is an exponent?If “x” is an exponent?

22

3 4x

33

loglogbby is read as “log base b of y”y is read as “log base b of y”

logbxix yff by

Definition of LogarithmDefinition of Logarithm

44

ExamplesExamples

1. log1. log339 = 29 = 2

2. log2. log881 = 01 = 0

3. log3. log55 (1/25) = -2 (1/25) = -2

1. 31. 322 = 9 = 9

2. 82. 800 = 1 = 1

3. 53. 5-2-2 = 1/25 = 1/25

Rewrite the equations in Rewrite the equations in exponential form.exponential form.

Logarithmic FunctionLogarithmic Function Exponential FunctionExponential Function

55

ExamplesExamples Evaluate the expressions.Evaluate the expressions.

Hint:Hint: For log For logbby ask yourself what power y ask yourself what power of of bb gives you gives you yy??

4. log464What power of 4 gives you 64?4x = 64Answer: 3 5. log20.125

What power of 2 gives you 0.125?2x = 0.125Answer: -3

6. log1/4256What power of ¼ gives you 256?1/4x = 256Answer: -4 7. log322

What power of 32 gives you 2?32x = 2Answer: 1/5

66

Common and Natural LogsCommon and Natural Logs

Common LogarithmCommon Logarithm(the base of 10 is not written)(the base of 10 is not written)

loglog1010x = log xx = log x

Natural Logarithm Natural Logarithm (remember “e” = natural base)(remember “e” = natural base)

loglogeex = ln xx = ln x

77

ExamplesExamples

Evaluate:Evaluate: (Round to 3 decimals)(Round to 3 decimals)

8. log 78. log 7= = 0.8450.845

9. ln 0.259. ln 0.25

= = -1.386-1.386

On TI-83:

LOG button is base 10 and is to the left of 7

LN button is base e and is to the left of 4

88

Logarithm Inverse PropertiesLogarithm Inverse Properties

( )( ) log xb a f xndg x bx

are of eachinverses other!

This means that...log

blog bx xan bdx xb

99

ExamplesExamples Simplify the expressions.Simplify the expressions.

= x

35= log 5 x = 3x

20log x10. 20

513. log 125x

blog xb xlogb xb x

log 512. 10 5

2411. log 4 2

1010

Finding Finding InversesInverses of Logarithms of Logarithms

SAME Steps as Before!!!SAME Steps as Before!!!

First, switch the First, switch the xx’s and ’s and yy’s.’s. Rewrite the logarithm equation as an Rewrite the logarithm equation as an

exponential equation.exponential equation. Solve for y.Solve for y.

1111

ExamplesExamples

Find the inverse of the following Find the inverse of the following functions.functions.

14. 14. y = logy = log88 x x

x = logx = log88yy

88xx = y = y

y = 8y = 8xx

Switch x and y

Re expwrite as onential

1212

ExamplesExamples

15. 15. y = ln (x – 10)y = ln (x – 10)

x = ln(y – 10)x = ln(y – 10)

eexx = y – 10 = y – 10

y = ey = exx + 10 + 10

Switch x and y

Re

exp

write as

onential

1313

316. ( ) log ( 1)f x x

3log ( 1)y x

3log ( 1)x y 3 1x y

3 1x y 1 ( ) 3 1xf x

Switch x and y

Re

exp

write as

onential

1414

HomeworkHomework p.490p.490

#16-64 evens#16-64 evens

1515

Algebra 2: Section 8.4Algebra 2: Section 8.4

Logarithmic FunctionsLogarithmic Functions(Day 2)(Day 2)

(Graphing…yeah!)(Graphing…yeah!)

1616

logbxix yff by

Definition of Logarithm Definition of Logarithm (Reminder)(Reminder)

SAME ASlogb

xix y ff b y

1717

Change of Base FormulaChange of Base Formula

Used to evaluate logs that are bases other Used to evaluate logs that are bases other than 10 or than 10 or ee..

Or to punch logs of base other than 10 or Or to punch logs of base other than 10 or ee into the calculator into the calculator (for graphing).(for graphing).

log lnlog

log lnc

u uu or

c c

loglog

logc

xx

c

Graphs of Logarithmic FunctionsGraphs of Logarithmic Functions8

6

4

2

-2

-4

-6

-10 -5 5 10

" "

!

SAME AS e graphs

except everything

is rotated

1919

Graphs of Logarithmic FunctionsGraphs of Logarithmic Functions

y = logy = logbb(x – h) + k(x – h) + k

Asymptote: Asymptote: hh ((x = “h”x = “h”))Domain:Domain: Range:Range:If b>1, curve opens upIf b>1, curve opens upIf 0<b<1, curve opens downIf 0<b<1, curve opens down

To graph: Show the asymptote Plot the x-intercept (calc or…..)

• find by setting y = 0 (will have to do for SEVERAL!!!)• rewrite as an exponential equation• Solve for x

( , )h ( , )

How to write in CalculatorHow to write in Calculator

2020

2

3

ln

log 3

l

log

og

ln

log 5

2

4

3

1

6

x

x

x

x

x

x

log( )4

(2)

x

log

ln 6x

log( ) 3x

ln( ) 3x

log 5x log 2

1log(3)

x

2121

ExamplesExamples State the asymptote, the domain, State the asymptote, the domain,

and the range of each function.and the range of each function.

1. 1. y = logy = log1/21/2x + 4x + 4

1/ 2log ( 0) 4y x

: 0asymptote x int : (16,0)x

: (0, ) ; ( , )D R

Curve opens down log

41

log2

xy

Graph and label

Asymptote!!!

?Does the graph disappear to

0x

2222

ExamplesExamples

2. y = log2. y = log33(x – 2)(x – 2)

3log ( 2) 0y x

: 2asymptote x int : (3,0)x

: (2, ) ; ( , )D R

Curve opens up

log( 2)0

log3

xy

2?Does the graph disappear at x

Graph and label

Asymptote!!!

2x

2323

HomeworkHomework p.491p.491

#65-76 all#65-76 all

State asymptote, x-intercept, domain, rangeState asymptote, x-intercept, domain, range Be sure asymptotes are graphed and labeledBe sure asymptotes are graphed and labeled