SOLVING EQUATIONS WITH VARIABLE EXPONENTS

Example 1 :

If 3^a+1 = 3^-a+7, what is the value of a ?

Solution :

3^a+1 = 3^-a+7

On both sides of the equal sign, we have same base. So, equate the powers.

a+1 = -a+7

Add a and subtract 1 on both sides, we get

a+a = 7-1

2a = 6

a = 3

Example 2 :

If 3^x+2 = y, then what is the value of 3^x in terms of y ?

(a) y+9  (b) y-9  (c) y/3  (d)  y/9

Solution :

3^x+2 = y

Using the property a^m+n = a^m ⋅ aⁿ

3^x ⋅ 3² = y

3^x ⋅ 9 = y

Dividing by 9 on both sides, we get

3^x = y/9

Example 3 :

If 2a-b = 4, what is the value of 4^a/2^b

Solution :

Given :

2a-b = 4

From 4^a/2^b,

4 = 2 ⋅ 2 ==> 2²

4^a/2^b = (2²)^a/2^b

We have power raised to another power. So, multiply the powers.

= 2^2a/2^b

= 2^2a-b

By applying the value of 2a-b = 4, we will get

= 2⁴

= 16

Example 4 :

If 2^x+3 - 2^x = k(2^x), what is the value of k ?

Solution :

2^x+3 - 2^x = k(2^x)

Using the property a^m+n = a^m ⋅ aⁿ

2^x+3 - 2^x = k(2^x)

2^x ⋅ 2³- 2^x = k(2^x)

Factoring 2^x,

2^x (2³- 1) = k(2^x)

Divide by 2^x, we get

(2³- 1) = k

k = 8-1

k = 7

Example 5 :

If 

√(x√x) = x^a

then what is the value of a ?

Solution :

√(x√x) = x^a

We can write √x as x^1/2

(x√x)^1/2 = x^a

(x√x)^1/2 = x^a

Take squares on both sides, we get

(x√x) = (x^a)²

(x√x) = x^2a

(x ⋅ x^1/2) = x^2a

x^1+1/2 = x^2a

x^3/2 = x^2a

2a = 3/2

a = 3/4

Example 6 :

If n³ = x and n⁴ = 20x, where n > 0, what is the value of x ?

Solution :

n³ = x ----(1)

n⁴ = 20x ----(2)

Applying the value of x in (1), we get

n⁴ = 20n³

n = 20

By applying the value of x in (1), we get

x = 20³

x = 8000

Example 7 :

Solve for x, 2(4^2x+1) = 128

Solution :

2(4^2x+1) = 128

Divide by 2 on both sides.

4^2x+1 = 64

64 = 4 ⋅ 4 ⋅ 4 ==>  4³

4^2x+1 = 4³

Bases are equal, so equate the powers.

2x+1 = 3

2x = 2

x = 1

Example 8 :

12^2x-1 = (∜12)^x

Solution :

12^2x-1 = (∜12)^x

12^2x-1 = ((12)^1/4)^x

12^2x-1 = (12)^x/4

2x-1 = x/4

4(2x-1) = x

8x-4 = x

8x-x = 4

x = 4/7

Example 9 :

3^4x+3 = 81^x

Solution :

3^4x+3 = 81^x

81 = 3⁴

3^4x+3 = (3⁴)^x

3^4x+3 = 3^4x

4x+3 = 4x

So, there is no solution.

Example 10 :

12 ⋅ 2^x-7 = 24

Solution :

12 ⋅ 2^x-7 = 24

Divide by 12 on both sides, we get

2^x-7 = 2¹

x-7 = 1

x = 1+7

x = 8