SOLVING EXPONENTIAL EQUATIONS WORKSHEET WITHOUT USING LOGARITHMS

Solve for x

Problem 1 :

2^x  =  32

Problem 2 :

3^x-2  =  1/9

Problem 3 :

Solve for x

6 ⋅ 3^x  =  54

Problem 4 :

Solve for x

4^x-1  =  (1/2)^1-3x

Problem 5 :

Solve for x

2^x+2  =  (1/4)

Problem 6 :

Solve for x

3^1-2x  =  1/27

Problem 7 :

Solve for x

5 ⋅ 2^x  =  40

Problem 8 :

Solve for x

9^x  =  7(3^x) + 18

Problem 9 :

Solve for x

3^(x+1) ⋅ 9^-x  =  (1/3)^x+1

Problem 10 :

Solve for x

2^x ⋅ 4^2-x  =  8

Problem 11 :

Solve for x

2^(x-1)⋅ 4^(2x+1)  =  32

Problem 12 :

Solve for x

5^3x-2  =  125^2x

Problem 13 :

A bread dough doubles in size every hour. You begin measuring the volume of the dough 1 hour after the dough is prepared. The volume y (in cubic inches) of the dough x hours after the dough is prepared is represented by 

y = 35(2^x-1)

When will the volume of the dough be 17920 cubic inches?

Problem 14 :

You deposit $750 in a savings account that earns 4% annual interest compounded yearly. Write and solve an exponential equation to determine when the balance of the account will be $1000

Problem 15 :

If 3^x = 5^y = 75^z, then

a) x + y - z = 0     b)  (2/x) + (1/y) = (1/z)

c)  1/x + 2/y = 1/z       d)  2/x + 1/z = 1/y

(1)  Solution : 

2^x  =  32 ----(1)

32  =  2⋅2⋅2⋅2⋅2

32  =  2⁵

From (1)

2^x  =   2⁵

By equating powers, we get

x  =  5

So, the value of x is 5.

(2)  Solution : 

3^x-2  =  1/9 ----(1)

9  =  3⋅3

9  =  3²

From (1)

3^x-2  =  1/3²

3^x-2  =  3^-2

By equating powers, we get

x-2  =  -2

x  =  -2+2

x  =  0

So, the value of x is 0.

(3)  Solution : 

6 ⋅ 3^x  =  54

Divide by 6 on both sides.

3^x  =  54/6

3^x  =  9

3^x  =  3²

By equating powers, we get

x  =  2

So, the value of x is 2.

(4)  Solution : 

4^x-1  =  (1/2)^1-3x

2^2(x-1)  =  2^-1(1-3x)

2^2x-2  =  2^-1+3x

By equating powers, we get

2x-2  =  -1+3x

2x-3x  =  -1+2

-x  =  1

x  =  -1

So, the value of x is -1.

(5)  Solution : 

2^x+2  =  (1/4)

2^x+2  =  (1/2²)

2^x+2  =  2^-2

By equating powers, we get

x+2  =  -2

x  =  -2-2

x  =  -4

So, the value of x is -4.

(6)  Solution : 

3^1-2x  =  1/27

3^1-2x  =  1/3³

3^1-2x  =  3^-3

By equating powers, we get

1-2x  =  -3

-2x  =  -3-1

-2x  =  -4

x  =  4/2

x  =  2

So, the value of x is 2.

(7)  Solution : 

5 ⋅ 2^x  =  40

Divide by 5 on both sides.

2^x  =  40/5

2^x  =  8

2^x  =  2³

By equating the power, we get 

x  =  3

So, the value of x is 3.

(8)  Solution : 

9^x  =  7(3^x) + 18

(3²)^x  =  7(3^x) + 18

(3^x)²  =  7(3^x) + 18

(3^x)² - 7(3^x) - 18  =  0

Let 3^x  =  a

a² - 7a - 18  =  0

(a-9)(a+2)  =  0

(9)  Solution : 

3^(x+1) ⋅ 9^-x  =  (1/3)^x+1

3^(x+1) ⋅ (3²)^-x  =  (3^-1)^x+1

3^(x+1) ⋅ 3^-2x  =  3^-1(x+1)

3^(x+1)-2x  =  3^(-x-1)

Equating powers

x+1-2x  =  -x-1

-x+1  = -x-1

-x+x  =  -1-1

So, there is no solution.

(10)  Solution : 

2^x ⋅ 4^2-x  =  8

2⋅ (2²)^2-x  =  2³

2¹⋅ 2^4-2x  =  2³

2^1+4-2x  =  2³

2^5-2x  =  2³

By equating powers, we get

5-2x  =  3

2x  =  5-3

2x  =  2

x  =  1

So, the value of x is 1.

(11)  Solution :

2^(x-1)⋅ 4^(2x+1)  =  32

2^(x-1)⋅ 2^2(2x+1)  =  2⁵

2^{(x-1)+2(2x+1)}  =  2⁵

2^x-1+4x+2  =  2⁵

2^5x+1  =  2⁵

By equating powers, we get

5x+1  =  5

5x  =  4

x  =  4/5

(12)  Solution :

5^3x-2  =  125^2x

5^3x-2  =  (5³)^2x

5^3x-2  =  5^6x

By equating powers, we get

3x-2  =  6x

6x-3x  =  -2

3x  =  -2

x  =  -2/3

So, the value of x is -2/3.

13) Solution :

y = 35(2^x-1)

When y = 17920

17920 = 35(2^x-1)

2^x-1 = 17920/35

2^x-1 = 512

2^x-1 = 2⁹

x - 1 = 9

x = 10

14)  Solution :

Initial deposit = $750

Compounding yearly 

A = P(1 + r/n)^t

r = 4% and n = 1 (Compounding yearly)

Balance = 1000

1000 = 750(1 + 0.04/1)^t

1000 = 750(1 + 0.04)^t

1000/750 = (1.04)^t

1.33 = (1.04)^t

t = 7.2

15) Solution :

3^x = 5^y = 75^z

Let 3^x = 5^y = 75^z = k

Each term can be equated to k

3^x = k, 5^y = k, 75^z = k

3 = k^1/x , 5 = k^1/y, 75 =  k^1/z

75 = k^1/z

3 x 5 x 5 = k^1/z

3 x 5² = k^1/z

Applying the value of 3 and 5, we get

k^1/x ⋅ (k^1/z)² = k^1/z

k^1/x ⋅ k^2/z = k^1/z

k^{(1/x) + (2/z)}  = k^1/z

1/x + 2/z = 1/z

So, option C is correct.

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