1

IntegratedIntegratedMathematicsMathematics

Solving Exponential Solving Exponential EquationsEquations

2

422 x162 :for which Find xx

4x

yx aa For yx

3

35 2)2( x

832 for which Find xx

5

3

35

x

x

Convert to thesame base

Equate the indices

4

)3(34 33 xx

34 273 for which Find xxx

9

934

x

xx

Convert to thesame base

Equate the indices

5

Solving Exponential EquationsSolving Exponential Equations

Product/DivisionProduct/Division

Exponential Exponential EquationsEquations

Addition/SubtractionAddition/Subtraction

ExponentialExponential

EquationsEquations

6

3223

3)1(23

555

5

155

xx

xx

125

1255 for which Find 13 xxx

Apply Law

Equate the indices

Convert base toits prime factors

3)22(3 55 xx

1

32

3)22(3

x

x

xx

Product/DivisionProduct/Division

Exponential EquationsExponential Equations

7

632 22 xx

634 for which Find 2 xxx

2

1

12

x

xApply Law

Equate the indices

12 66 x

Convert base toits prime factors

Product/DivisionProduct/Division

Exponential EquationsExponential Equations

8

Solving Addition/SubtractionSolving Addition/Subtraction

Exponential EquationsExponential Equations

Example: Example:

Apply Law

Let y be thecommon term

Convert base toits prime factors

Solve for y

Solve for x

Equate Indicesor Bases

Solve for x

SubstitutionMethod

033 )1(252

xx

9

033

033225

)1(25

2

2

xx

xx

093 for which Find 152

xxx

225 332 xx

Addition/SubtractionAddition/Subtraction

Exponential EquationsExponential Equations

Apply Law

Equate Indices

Convert base toits prime factors

Solve for x

1or 3

0)1)(3(

032

2252

2

xx

xx

xx

xx

Check your answer:

0

)3(3

93

93

224

24

1359

Check your answer:

0

)3(3

93

93

224

24

1151

10

055 )2(232

xx

0255 for which Find 232

xxx

Addition/SubtractionAddition/Subtraction

Exponential EquationsExponential Equations

Apply Law

Equate Indices

Convert base toits prime factors

Solve for x

)2(23 552 xx

3or 1

0)3)(1(

034

432

2

xx

xx

xx

xx

Check your answer:

0

55

255)2(24

231

Check your answer:

0

55

255)6(239

639

11

72222 3 xx

2722 for which Find 3 xxx

72)2(82 xx

8

729

728

y

y

yy

Addition/SubtractionAddition/Subtraction

Exponential EquationsExponential Equations

Apply Law

Let y be thecommon term

(Use a substitution)

Separate the Indices

xy 2Let 3

22

28

,2 From

3

x

y

x

x

x

Solve for y

Solve for x

72)2(82 xxy y Check your answer:

72

648

22 333

12

2022 2)1(2 xx

2024 for which Find 21 xxx

20222

2022222

222

xx

xx

4

205

y

y

Addition/SubtractionAddition/Subtraction

Exponential EquationsExponential Equations

Apply Law

Use a substitution

Separate the Indices

xy 22Let 1

22

24

,2 From

22

2

2

x

y

x

x

x

Solve for y

Solve for x

202)2(4 22 xxy y Check your answer:

20

22

2224

)1(2)11(2

13

3

1333 for which t Find 212 tt

Addition/SubtractionAddition/Subtraction

Exponential EquationsExponential Equations

Apply Law

Use a substitution

Separate the Indices

ty 3Let

Solve for y

Solve for t

3

13)3(3)(33 221 tt

3

13)3(3)3(3 22 tt

3

13)3(3)3(3 22 tty y

010279y

3

1093y

2

2

y

y

3

10

3

1

0)103)(13(

yy

yy

1

33

33

1

3

1

t

y

t

t

t

Soln) (No

33

10 t

14

01292 for which x Find 212 xx

Addition/SubtractionAddition/Subtraction

Exponential EquationsExponential Equations

Apply Law

Use a substitution

Separate the Indices

Solve for y

Solve for x

01)2)(2(92)(2 212 xx

01)2)(4

1(9)(2

2

1 2 xx

01)2)(4

9()(2

2

1 2 xx

xy 2Let

y y

4 2

1

0)4)(12(

0492 2

yy

yy

yy

1

22

22

1

2

1

x

y

x

x

x

2

22

24

2

2

x

y

x

x

x

2or 1 :Ans xx

15

Summary Exponential EquationsSummary Exponential Equations

Product/DivisionProduct/Division

Exponential EquationsExponential Equations

Addition/SubtractionAddition/Subtraction

Exponential EquationsExponential Equations

Apply Law

Equate the indices

Convert base toits prime factors

Apply Law

Let y be thecommon term

Convert base toits prime factors

Solve for y

Solve for x

Equate Indicesor Bases

Solve for x