Activity 6: Solving Exponential Equations - mrsk.ca

MHF4U_2011: Advanced Functions, Grade 12, University Preparation

Unit 3: Exponential and Logarithmic Functions

Activity 6: Solving Exponential Equations

Homework Assignment

1. Show two ways of solving the problem below.

3^{x + 2} - 3^x = 216

2. Solve the following exponential equations using a common base.

a) 4^x = 16^{(x - 2)} b) 6^2x = 216^{(3x - 5)} d) (5)(5)^x+2 = 25^2x e) 3^{(x + 3)} - 3^x = 78

3. Solve the following exponential equations using logarithms.

a) 3^x = 9^x+3 b) 75(1.08)ⁿ = 450 c) 48(1.03)ⁿ = 96 d) 2^x = 3^x+1

Homework Assignment SOLUTIONS

1. Show two ways of solving the problem below.

2

3^x+ −3^x =216

Using a common base:

( )

2 2

3

3 3 216

3 3 1 216

3 216 8 3 27

3 3

3

x x

x

x

x x

x

+ − =

− =

=

=

=

=

Or using logarithms:

( )

2 2

3 3 216

3 3 1 216

3 216 8 3 27

log 3 log 27 log 3 log 27

log 27 log 3 3

x x

x

x

x x

x x x

+ − =

− =

=

=

=

=

=

=

2. Solve the following exponential equations using a common base.

a) 4^x = 16^{(x - 2)}

( )

^{2 2}

2 4

4 4

4 4 2 4

4 4

x x

x x

x x

x x

−

−

=

=

= −

− = −

=

b) 6^2x = 216^{(3x - 5)}

( )

2 (3 5)

3 5

2 3

2 9 15

6 216

6 6

6 6

2 9 15

7 15

15 7

x x

x x

x x

x x

x x

−

−

−

=

=

=

= −

− = −

=

( )

^{1 4 1}

(

²

)

^{3 3}

4 1 6 6

2 2

2 2

4 1 6 6

2 5

5 2

x x

x x

x x

x x

+ +

− −

− − − −

=

=

− − = − −

= −

= − d) (5)(5)^x+2 = 25^2x

( ) ( )

( )

1 2 2

3 2 2

3 4

5 5 25

5 5

5 5

4 4

3 4

4 3

x x

x x

x x

x x

x x

+

+ +

=

=

= + =

− = −

=

e) 3^{(x + 3)} - 3^x = 78

( )

( )

( 3) 3

3 3 78

3 3 1 78 3 26 78 3 3

1

x x

x x x

x

+ − =

− =

=

=

=

3. Solve the following exponential equations using logarithms.

a) 3^x = 9^x+3

( )

( )

3

3

3 9

log 3 log 9

log 3 3 log 9 log 3 log 9 3log 9 log 3 log 9 3log 9 log 3 log 9 3log 9

3log 9 log 3 log 9

6

x x

x x

x x

x x

x x

x x x

+

+

=

=

= +

= +

− =

− =

= −

= −

b) 75(1.08)ⁿ = 450

( )

75 1.08 450 1.08 6

log1.08 log 6 log1.08 log 6

log 6 log1.08 23.28

n

n n

n n n

=

=

=

=

=

c) 48(1.03)ⁿ = 96

( )

48 1.03 96 1.03 2

log1.03 log 2 log1.03 log 2

log 2 log1.03 23.45

n

n n

n n n

=

=

=

=

=

d) 2^x = 3^x+1

( )

( )

1 1

2 3

log 2 log 3

log 2 1 log 3 log 2 log 3 log 3 log 2 log 3 log 3 log 2 log 3 log 3

log 3 log 2 log 3

2.71

x x

x x

x x

x x

x x

x x x

+

+

=

=

= +

= +

− =

− =

= −

−