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