## Exponential Equation

An equation which contains the unknown variable appearing as an exponent of a base is known as an exponential equation. In the equation 5x = 25, the unknown variable x is an exponent of base 5. So this equation is called an exponential equation.

The axioms given below help to solve the exponential equations:

a)    If xa = ya, then x = y

b)   If xa=xb, then a = b

c)    If xa = 1, then xa = x0. So a = 0

The following steps are useful to solve the exponential equation:

1.    Simplify both sides of the equation

2.    Make both sides of the equation into the same base.

3.    Equate their exponents and simplify.

### Workout Examples

Solution: Here,

a)      2x+4 = 8x

or,     2x+4 = (23)x

or,     2x+4 = 23x

x+4 = 3x

or,     4 = 3x – x

or,     4 = 2x

or,     x = 4/2

or,     x = 2

Example 3: Solve: 32x+1 – 9x+1 + 54 = 0

Solution: Here,

32x+1 – 9x+1 + 54 = 0

or,     32x.31 – 9x.91 = -54

or,     9x.3 – 9x.9 = -54

or,     9x(3 – 9) = -54

or,     9x.-6 = -54

or,     9x = -54/-6

or,     9x = 9

or,     9x = 91

x = 1 or,     25x2 + 25 = 626x

or,     25x2 – 626x + 25 = 0

or,     25x2 – 625x – x + 25 = 0

or,     25x(x – 25) – 1(x – 25) = 0

or,     (x – 25)(25x – 1) = 0

Either, x – 25 = 0

or, x = 25

or, 5a = 52

a = 2

Or, 25x – 1 = 0

or, 25x = 1

or, x = 1/25

or, x = 1/52

or, 5a = 5-2

a = -2

a = 2 or -2

Example 7: Solve: 4 × 3x+1 = 27 + 9x

Solution: Here,

4 × 3x+1 = 27 + 9x

or,     9x – 4 × 3x+1 + 27 = 0

or,     32x – 4 . 3x . 31 + 27 = 0

or,     (3x)2 – 12 . 3x + 27 = 0

Let, 3x = a

Then, a2 – 12a + 27 = 0

or,     a2 – 9a – 3a + 27 = 0

or,     a(a – 9) -3(a – 9) = 0

or,     (a – 9)(a – 3) = 0

Either, a – 9 = 0

or, a = 9

or, 3x = 32

x = 2

Or, a – 3 = 0

or, a = 3

or, 3x = 31

x = 1

x = 1 or 2

You can comment your questions or problems regarding the exponential equations here.