CP Math Lesson 10-3 Inverses of Logarithmic and Exponential functions

CP MathCP MathLesson 10-3Lesson 10-3Inverses of Logarithmic and Inverses of Logarithmic and Exponential functions. Exponential functions.

Finding the Inverse of a Finding the Inverse of a function.function.

y = x – 2 y = x – 2

2yx

yx 2

Exchange ‘x’ and ‘y’Exchange ‘x’ and ‘y’

Add ‘2’ (left and right)Add ‘2’ (left and right)

is the inverse of: y = x - 2 is the inverse of: y = x - 2 2xy

Solve for ‘y’Solve for ‘y’

ExtremelyExtremely important idea. important idea.

xxf 3)( Find the inverse of f(x)Find the inverse of f(x)


xx 3log3

Power base 3 Power base 3 log base 3 log base 3 are inverse functionsare inverse functions

““undo” power base 3 undo” power base 3 log base 3 log base 3

yx 3loglog 33

?)(1 xf

xy 3 yx 3

xy 3log

yx 3


32)( xxf Find the inverse of f(x)Find the inverse of f(x)


xx 2log2


““undo” power base 2 undo” power base 2 log base 2 log base 2

322 2loglog yx

?)(1 xf

32 xy 32 yx

3log2 yx

32 yx

yx 2log3

xy 2log3

Exchange ‘x’ and ‘y’ Exchange ‘x’ and ‘y’ 1)4(3)( xxf

How do you find an inverse function?How do you find an inverse function?

Exchange ‘x’ and ‘y’ then solve for ‘y’Exchange ‘x’ and ‘y’ then solve for ‘y’

1)4(3 yx Divide left/right by 3Divide left/right by 3

1)4(3

yx““undo” power base 4undo” power base 4

144 )4(log

3log yx

13

log4 yx

Add ‘1’ left/rightAdd ‘1’ left/right

3log1 4

xy

3log1)( 4

1 xxf

?)(1 xf

Your Turn: Your Turn: Find for each of the following” Find for each of the following”

15.15. xxf )3(2)( )(1 xf

xy )3(2

yx)3(

2

yx)3(log

2log 33

yx

2log3

yx )3(2

2log3

xy

Your Turn: Your Turn: FindFind

16.16.

)(1 xf

134 xy

13log4 yx

yx 3log1 4 y

x

3

log1 4

134)( xxf

134 yx

xy 4log31

31

Your Turn: Your Turn: FindFind

17.17.12)5(4)( xxf

)(1 xf

12)5(4

yx

1255 )5(log

4log

yx

12)5(4 xy12)5(4 yx

124

log5

yx

yx

24

log1 5

y

x

24

log1 5

yx

4log

2

1

2

15


xxf 3log)( Find the inverse of f(x)Find the inverse of f(x)


xx 3log3


yx 3log

Isolate the log, undo the logIsolate the log, undo the logyx 3log33

““undo” log base 3 undo” log base 3 power base 3 power base 3yx 3

)(1 xf

xy 3


)1(log2)( 3 xxf Find the inverse of f(x)Find the inverse of f(x)


xx 3log3


)1(log2 3 yx

Isolate the log, undo the logIsolate the log, undo the log

Divide left/right by 2Divide left/right by 2

)1(log2 3 yx

““undo” log base 3 undo” log base 3 power base 3 power base 3)1(log2 333 yx

132 yx

Add ‘1’, Add ‘1’, 132 x

y

)(1 xf

Your Turn: Your Turn:

18.18.

19.19. )3ln(2)( xxf

xxf 2log)(

20.20.

Find for each of the following” Find for each of the following” )(1 xf

5

)4(log)( 5

xxf

Simplifying Logarithmic Simplifying Logarithmic ExpressionsExpressions using inverse functions using inverse functions

xb xb log ?7 10log7

?ln 4 xe

Your turn:Your turn:

21.21. ?2 5log2

?4log 104

22.22.

10 10

Solve equations using Solve equations using inverse functions.inverse functions.

““isolate the square, isolate the square, undo undo the the square”square”

8)4(2 2 x

4)4( 2 x÷2÷2 ÷2÷2

4)4( 2 x

Isolate the base and itsIsolate the base and its exponent.exponent.

““undo” squaringundo” squaring

Use the Use the inverseinverse of the of the square function.square function.

24 x Even Root Even Root !!!!!! ± ± resultresult

24 x

x = 6, 2x = 6, 2

Do the mathDo the math!!!!!!

24 ,24 x

Key point: Key point: To solve an equation you must To solve an equation you must know how to “undo” a function.know how to “undo” a function.

8)4(2 2 xTo solve this you must “undo” a square function.To solve this you must “undo” a square function.

84 xTo solve this you must “undo” a square root function.To solve this you must “undo” a square root function.

642 35

xTo solve this you must “undo” a 5/2 power.To solve this you must “undo” a 5/2 power.

Your turn:Your turn:Solve each of the following:Solve each of the following:

1.1. 38 x

2.2. 52 x

3.3. 32

4 x

3 33 8 x 2x

5552 x 32x

23

32

23

4

x 8x

How do you solve the How do you solve the following function for ‘x’?following function for ‘x’?

153 x

Logarithm Logarithm FunctionFunction

xxf 3log)(

Exponential Exponential FunctionFunction

xxf 3)( reflections of each otherreflections of each other

across the line y = x.across the line y = x.

They are inverses They are inverses of each other.of each other.

How do you solve the following How do you solve the following function for ‘x’?function for ‘x’?

153 x

15log3log 33 x

xxf 3)( xxf 3log)(

These are inverse functions of each These are inverse functions of each other.other.

You must You must “undo” “undo” a power a power base 3.base 3.

Log base 3 Log base 3 “un-does” “un-does” power power base 3 !!!!base 3 !!!!

15log3x


xx 3log3

15log3log 33 x


15log3x

Solve for ‘x’Solve for ‘x’ 112 x

11log2log 22 x

11log2x

Log base 2 is the inverse of power Log base 2 is the inverse of power base 2 !!!!base 2 !!!!

Your turn:Your turn:““undo” each of the following to undo” each of the following to solve for x:solve for x:

4.4. 222 x

5.5. 154 x

6.6. 10xe

22log2log 22 x 22log2x


10lnln xe 10lnx

Change-of-Base Formula for Change-of-Base Formula for Logarithms Logarithms

Change of Base FormulaChange of Base Formula: :

c

aa

b

bc log

loglog

Change to log Change to log base 10base 10 or or base ‘e’base ‘e’ (your calculator can do these). (your calculator can do these).

5log4

Convert to base 10.Convert to base 10.

4log

5log

10

106021.0

699.0 161.1

Another Example:Another Example:Calculate the following using the Calculate the following using the base conversion formula.base conversion formula.

?9log2 2log

9log

301.0

9542.0 17.3

?9log2 2ln

9ln

69315.0

19722.2 17.3

Solve for ‘x’Solve for ‘x’ 112 x

11log2log 22 x

11log2x

Log base 2 is the inverse of power Log base 2 is the inverse of power base 2 !!!!base 2 !!!!

?11log2 2log

11log

301.0

04.1 46.3

112 46.3 Check your solutionCheck your solution

Your turn: Your turn: Calculate ‘x’ using the Calculate ‘x’ using the base conversion formula.base conversion formula.

7.7.

8.8.

x7log8

x9log3

x8log

7log

10

10

x3log

9log

9358.0x

2x

Your turnYour turn..

125 x9.9.

Solve the following.Solve the following.

10.10. 227 x

x5log

12log

10

10 544.1x



x7log

22log

10

10 5885.1x

Solve for ‘x’Solve for ‘x’ 12)4(2 3 x

6log4log 43

4 x

6log3 4x

““Isolate the power, “undo” the Isolate the power, “undo” the power”power”

6)4( 3 x

Divide by 2Divide by 2

Undo power base 4 with log base 4Undo power base 4 with log base 4

Change of base formulaChange of base formula

4log

6log3 x

292.13 x Add 3Add 3 292.4x

Your turnYour turn..113 2 x

11.11.

Solve the following.Solve the following.

12.12. 199 3 x

11log3log 32

3 x 11log2 3x

3log

11log2 x 1827.22 x 1827.4x

19log9log 93

9 x 19log3 9x

9log

19log3 x 3401.13 x 6599.1x

Your turnYour turn..

1524 32 x13.13.

““isolate the power, “undo” the power”isolate the power, “undo” the power”

11log2log 232

2 x 11log32 2x

2log

11log32 x

112 32 x

4594.332 x

4594.62 x

2297.3x

Your turnYour turn..““isolate the power, “undo” the power”isolate the power, “undo” the power”

14.14. 2553 23 x

28log5log 523

5 x 28log23 5x

5log

28log23 x

285 23 x

0704.223 x

0704.03 x

0235.0x

Real World Logarithmic Real World Logarithmic ModelModel

65log93 ds

FormulaFormula relating distance (d) that a tornado travels relating distance (d) that a tornado travels and the wind speed (s) inside the cone of the tornado.and the wind speed (s) inside the cone of the tornado.

In 1925, a tornado traveled 220 miles through 3 states. In 1925, a tornado traveled 220 miles through 3 states. Estimate the Estimate the wind speedwind speed inside the tornado. inside the tornado.

(1) Write formula(1) Write formula

(2) Plug numbers into the formula(2) Plug numbers into the formula

65log93 ds

65220log93 s

(3) Solve for the unknown variable in the formula(3) Solve for the unknown variable in the formula

mphs 8.282

Your turn:Your turn:65log93 ds

FormulaFormula relating distance (d) that a tornado travels relating distance (d) that a tornado travels and the wind speed (s) inside the cone of the tornado.and the wind speed (s) inside the cone of the tornado.

23. 23. A tornado traveled 100 miles. Estimate the A tornado traveled 100 miles. Estimate the wind speed wind speed inside the tornado.inside the tornado.



65log93 ds

65100log93 s


mphs 251

Real World Logarithmic Real World Logarithmic ModelModel

65log93 ds

What if the problem wasWhat if the problem was::

“ “The wind speed is 200mph, The wind speed is 200mph, how farhow far will the tornado travel will the tornado travel on the ground?on the ground?



65log93 ds

65)log(93200 d


(remember: “isolate the log, undo the log”)(remember: “isolate the log, undo the log”)

Real World Logarithmic ModelReal World Logarithmic Model65log93 ds

““The wind speed is 200mph, how far will the tornado travel The wind speed is 200mph, how far will the tornado travel on the ground?on the ground?

Subtract 65Subtract 65 from both sides from both sides

)log(93135 d65)log(93200 d

Divide by 93Divide by 93 (both sides) (both sides)

remember: “remember: “isolate the radicalisolate the radical” ? ” ? same thing, “ same thing, “isolate the logisolate the log””

)log(4516.1 d We need the “We need the “inverse functioninverse function” of ” of log (base 10).log (base 10).

)log(4516.1 1010 d d4516.110 3.28

Exponential Exponential base 10base 10

We will solve log and exponential equations later.We will solve log and exponential equations later.

Simplifying Logarithmic Simplifying Logarithmic ExpressionsExpressions using inverse functions using inverse functions

?10 4log xb xb log ?2log 32

?27log3

?8log2

Can I rewrite 27 as a power of 3?Can I rewrite 27 as a power of 3?

Can I rewrite 8 as a power of 2?Can I rewrite 8 as a power of 2?

33 3log

32 2log

4 3

3

3

Your turn:Your turn:24.24.

25.25.

?16log 32

26.26.

?2log 32

?16

1log

52

Your turn:Your turn:Find the inverse function of:Find the inverse function of:

27.27.xy 10 28.28. )3ln( xy

Convert to Convert to exponential formexponential form

29.29. 264log x

What is the base?What is the base?

x7ln 5log x

30.30.

31.31. 32. 32.

Convert to Convert to logarithm formlogarithm form

43 x

Inverse function of a logarithmInverse function of a logarithm

Inverse of a log (base 4) is an exponential (base 4)Inverse of a log (base 4) is an exponential (base 4)

xxg 4)(

xxf 4log)( Inverse of a log is an Inverse of a log is an exponential function.exponential function.

Compose a function with its “Compose a function with its “inverse functioninverse function” to “undo” the” to “undo” the original function.original function.

x xxgf 4log))(( 4Log (base 4) of an exponential (base 4) Log (base 4) of an exponential (base 4)


Inverse of a log (base e) is an exponential (base e)Inverse of a log (base e) is an exponential (base e)

xexg )(

xxf ln)( Inverse of a log is an Inverse of a log is an exponential function.exponential function.


xxexgf ln))(( Log (base e) of an exponential (base e) Log (base e) of an exponential (base e)


Inverse of an exponential (base 4) is a log (base 4) Inverse of an exponential (base 4) is a log (base 4)

xxf 4)(

xxg 4log)(

Inverse of an Inverse of an exponential functionexponential function is a is a logarithm function.logarithm function.


x xxgf 4log4))(( Exponential (base 4) Exponential (base 4) raisedraised to the log (base 4) to the log (base 4)


Inverse of an exponential (base e) is a log (base e) Inverse of an exponential (base e) is a log (base e)

xexf )(

xxg ln)( Inverse of an Inverse of an exponential functionexponential function is a is a log function.log function.


xxexgf ln))(( exponential (base e) exponential (base e) raisedraised to the log (base e) to the log (base e)

Finding the Inverse FunctionFinding the Inverse FunctionHow do you find the inverse of:How do you find the inverse of:

(1) Exchange ‘x’ and ‘y’(1) Exchange ‘x’ and ‘y’

(2) Solve for ‘y’(2) Solve for ‘y’

How do you find the inverse of:How do you find the inverse of:xy 4

(1) Exchange ‘x’ and ‘y’(1) Exchange ‘x’ and ‘y’ yx 4

(2) Solve for ‘y’(2) Solve for ‘y’yx 4loglog 44

yx 4log

yx 7log

xy 7logxy 77 7log (compose with(compose with

its inverse function)its inverse function)xy 7

(compose with(compose with its inverse function)its inverse function)

Your Turn: Your Turn: What is the base?What is the base?

33.33.

34.34.

35.35.

x5ln

x20log

x8log2

37.37. x43

36.36. 5log2 x

Convert to an exponentialConvert to an exponential

Convert to a logarithmConvert to a logarithm

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CP Math Lesson 10-3 Inverses of Logarithmic and Exponential functions