SOLVING EXPONENTIAL EQUATIONS EXAMPLES

Example 1 :

Solve 2^3x-4 = 8^x-1

Solution :

4^3x-4 = 8^x-1

Trying to write the bases on both sides of the equal sign as same base.

4^3x-4 = 2^3(x-1)

2^2(3x-4) = 2^{3x - 3}

Now we get the same bases on both sides. So, equating the powers. We get,

6x - 8 = 3x - 3

6x - 3x = -3 + 8

3x = 5

x = 5/3

So, the value of x is 5/3.

Example 2 :

Solve 3^x-4 = 9√3

Solution :

3^x-4 = 9√3

3^x-4 = 3²√3

3^x-4 = 3^2+(1/2)

3^x-4 = 3^(5/2)

x - 4 = 5/2

x = 5/2 + 4

x = 13/2

So, the value of x is 13/2.

Example 3 :

Solve 9^x+2 = 27^x

Solution :

9^x+2 = 27^x

3^2(x+2) = 3^3x

3^{2x + 4} = 3^3x

Equating the powers, we get

2x + 4 = 3x

2x - 3x = -4

-x = -4

x = 4

So, the value of x is 4.

Example 4 :

Solve 216^x = 6^x+10

Solution :

216^x = 6^x+10

6^3x = 6^x+10

3x = x + 10

3x - x = 10

2x = 10

x = 10/2

x = 5

So, the value of x is 5.

Example 5 :

Solve 36^-3x+3 = (1/216)^x+10

Solution :

36^-3x+3 = (1/216)^x+10

6^2(-3x+3) = (1/6)^3(x+10)

6^2(-3x+3) = 6^-3(x+10)

Equating the powers, we get

2(-3x + 3) = -3(x + 10)

-6x + 18 = -3x - 30

-6x + 3x = -30 - 18

-3x = -48

x = 48/3

x = 16

So, the value of x is 16.

Example 6 :

Solve 9^2x = 27^x+4

Solution :

9^2x = 27^x+4

Writing 9 and 27 in exponential form with the same base.

(3²)^2x = (3³)^x+4

3^4x = 3^3x+12

4x = 3x + 12

4x - 3x = 12

x = 12

So, the value of x is 12.

Example 7 :

Solve 81^x = 243^x+2

Solution :

81^x = 243^x+2

(3⁴)^x = (3⁵)^x+2

3^4x = 3^5(x+2)

4x = 5x + 10

4x - 5x = 10

-x = 10

x = -10

So, the value of x is -10

Example 8 :

Solve for n

Solution :

17n + 8 = 4

17n = 4 - 8

17n = -4

n = -4/17

So, the value of n is -4/17.

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