SOLVING EXPONENTIAL EQUATIONS EXAMPLES

To solve exponential equations, we have to follow the steps given below.

Step 1 :

If we see more than one terms in any of the sides, using properties of exponents we have to convert them as one term.

Step 2 :

Try to write the bases on both sides of equal signs as  same base.

Step 3 :

  • If bases are equal, then we can equate the powers.
  • If the powers are equal, then we can equate the bases.

Example 1 :

Solve 23x-4 = 8x-1

Solution :

43x-4 = 8x-1

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

43x-4 = 23(x-1)

22(3x-4) = 23x - 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 3x-4 = 9√3

Solution :

3x-4 = 9√3

3x-4 = 32√3

3x-4 = 32+(1/2)

3x-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 9x+2 = 27x

Solution :

9x+2 = 27x

32(x+2) = 33x

32x + 4 = 33x

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 216x = 6x+10

Solution :

216x = 6x+10

63x = 6x+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

62(-3x+3) = (1/6)3(x+10)

62(-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 92x = 27x+4

Solution :

92x = 27x+4

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

(32)2x = (33)x+4

34x = 33x+12

4x = 3x + 12

4x - 3x = 12

x = 12

So, the value of x is 12.

Example 7 :

Solve 81x = 243x+2

Solution :

81x = 243x+2

(34)x = (35)x+2

34x = 35(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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