Continuing with Operation-Inverse Pairs Copyright 2014 Scott Storla.

Copyright 2014 Scott Storla

Continuing with Operation-Inverse Pairs


All our operations have an inverse. What the operation “does” the inverse operation “undoes”.

Logarithm Exponential

Square Square Root

Multiplication Division

Addition Subtraction


10

log 100

log 7x

2

logb x

x

10

b

y

2100

yb b

1010




Square Square Root






Square Square Root




Solving Equations using a Two Column Table

Procedure – Solving Equations Using a Two Column Table

1. Build a two-column table. Label the left column for operations and the right column for inverses.

2. Imagine substituting a value for the variable and make a row for each operation that “acts on” the variable.

3. In the left column list the operations, in order, that you would use to simplify the expression if the variable was a value.

4. In the right column list the inverse operations, row by row for the operations found in step 3.

5. Start at the bottom of the right hand column and work your way to the top. You are using a reverse order of operations to isolate the variable.

6. Check any solutions. You can always use the left hand column to help simplify the expression.


Convention for rounding while solving equations.

Hold off on rounding until the final step.


Solving logarithmic equations

Property – The Exponential Property of Equality

English: Raising both sides of an equation as the power of the same base results in an equivalent equation.

Example: log 4y is equivalent to log 410 10y

Algebra: If s t then s tb b .

Property – The Second Inverse Property of Logarithms

English: If a logarithm of base b is written as an exponent of the same base the expression simplifies to the argument of the logarithm.

Example: log810 8 or lnye y

Algebra: logb xb x b, 0x and 1b


Solve. Check your solution(s).

3 logy

3 log10 10 y

y

3 log 1,000

3 3

1,000


4 ln t

4 ln te e

0.01832 t

4 ln 0.01832

4 3.9998

4 ln t



2ln 3x

2ln 3xe e

1.9477x

2ln 1.9477 3

0.66665 0.6



log 0.06p

log 0.0610 10p

0.871p

log 0.871 0.06

0.05998 0.06



ln 24 26p

Solve using a two column table

ln 24 26p

ln 2p

ln 2pe e

7.3891p

ln 7.3891 24 26

2.000... 24 26

26.000... 26

Oper Inv

ln ^

24 24

e

24 24


log 0.02 0.005p

log 0.02 0.005p

log 0.025p

log 0.02510 10p

1.0593p log 1.0593 0.02 0.005

0.0250... 0.02 0.005

0.0050... 0.005

0.02 0.02 Oper Inv

log 10 ^

0.02 0.02



3ln 7 2t

3ln 7 2t

3 3

ln 3t

ln 3te e

7 7

3ln 9t

20.0855t

3ln 20.0855 7 2

3 2.999... 7 2

8.999... 7 2

1.999.. 2


Oper Inv

ln ^

3 3

7 7

e


1.5 1.2ln 0.25h


1.25 1.2ln h1.2 1.2

1.25/1.2 ln he e

1.5 1.2ln 2.8339 0.25

1.5 1.2 1.0417 0.25

1.5 1.2500... 0.25

1.5 1.500...

1.25ln

1.2h

2.8339 h

1.5 1.2ln 0.25h Oper Inv

ln ^

1.2 1.2

0.25 0.25

e

0.25 0.25




Square Square Root






Square Square Root




Solving square root equations

Property: The Squaring Property of Equality

English: When you square both sides of an equation the new equation will have all the solutions of the original equation.

Example: The solutions to 3x are included in the solution set

of 2 23x .

Algebra: All the solutions of x b are included in the solutions of

2 2x b . 0x

Note: The new solution set may also contain extraneous solutions which don’t make the original equation true.



4 p

224 p

16 p

4 16

4 4


1 3t

2 2

1 3t

1/ 9t

1 1

3 3


1 1

39

1 1

9 3



e x

22e x

2e x

2e e

e e


3 5k


3 5k

2 k

222 k

4 k

Oper Inv

^2

5 5

3 4 5

3 2 5

3 3

5 5


9 9

9 2 16p


9 2 16p

2p

4p

9 18p

2 2

2 22p

9 4 2 16

9 2 2 16

18 2 16

16 16

Oper Inv

^2

9 9

2 2


0.5 0.5

0.5 1.05 0.25w


2.56w

1.05 1.05

0.5 1.05 0.25

0.5 2.56 1.05 0.25

0.5 1.6 1.05 0.25

0.8 1.05 0.25

0.25 0.25

w

Oper Inv

^2

0.5 0.5

1.05 1.05

0.5 1.05 0.25w

0.5 0.8w

1.6w

2 2

1.6w


0.25 1.54t


Operations Inverses

^2

4 4

1.5 1.5

0.25 1.54t

1.5 1.5

1.25 4t

44 1.25 41

t

5 t

225 t

25 t

250.25 1.54

50.25 1.54

0.25 1.25 1.5

0.25 0.25




Square Square Root






Square Square Root




Solving quadratic equations

Property – The Square Root Property

English: If the square of a variable is equal to a constant then the variable itself is equal to the square root of the constant or its opposite.

Example: If 2 4x then 4 or 4x x .

Algebra: If 2x a then or x a x a . 0a .

Note: The idea that x equals or a a is often written x a .

Property – The Inverse Property of Squaring and Square Root

English: For nonnegative radicands taking a square root and squaring are inverse operations.

Example:

24 4

Algebra: 2 , 0x x x


Solve

2 25y

5y

5,5

25 25

25 25

5y

25 25

25 25

5y

25y



2 12x

12x

2 3x

12 12

4 3 12

222 3 12

22 3 12



22

17k

2

17k

2 2

17 17

2

22 21

17 17

22 2

17 17


2 1 10x

2 1 10x

3x

2 9x

9x


Oper Inv

^2

1 1

1 1

23 1 10

9 1 10

10 10


2 72x

Oper Inv

^2

6 6

2 2

72x

2 2 2 3 3x


4 2 9x

2 3 2x

6 2x

26 22 14

6

226 22 14

6

36 22 14

6 2 12 14

14 14

22 14

6

x

2 2

212

6

x 6 6

722 14

6


2 4

5x

Oper Inv

^2

5 5

9.5 9.5

4

5x

2

5x


22 5

5 9.5 5.55

22

2

2 55 9.5 5.5

5

4 55 9.5 5.5

25

1009.5 5.5

25

4 9.5 5.5

5.5 5.5

25 9.5 5.5x 9.5 9.5

5 5

25 4x

2 5

5x

2 5

5 5x

4

5x

25 9.5 5.5x


Continuing with Operation-Inverse Pairs

Date post:	24-Dec-2015
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Continuing with Operation-Inverse Pairs Copyright 2014 Scott Storla.

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