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 5^x = 25, the unknown variable x is an exponent of base 5. So this equation is called an exponential equation.

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The axioms given below help to solve the exponential equations:

a)    If x^a = y^a, then x = y

b)   If x^a=x^b, then a = b

c)    If x^a = 1, then x^a = 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)      2^x+4 = 8^x

          or,     2^x+4 = (2³)^x

          or,     2^x+4 = 2^3x

          ∴        x+4 = 3x

          or,     4 = 3x – x

          or,     4 = 2x

          or,     x = 4/2

          or,     x = 2 

      

           

Example 3: Solve: 3^2x+1 – 9^x+1 + 54 = 0

Solution: Here,

                   3^2x+1 – 9^x+1 + 54 = 0

          or,     3^2x.3¹ – 9^x.9¹ = -54

          or,     9^x.3 – 9^x.9 = -54

          or,     9^x(3 – 9) = -54

          or,     9^x.-6 = -54

          or,     9^x = -54/-6

          or,     9^x = 9

          or,     9^x = 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, 5^a = 52

          ∴ a = 2

Or, 25x – 1 = 0

          or, 25x = 1

          or, x = 1/25

          or, x = 1/5²

          or, 5^a = 5^-2

          ∴ a = -2

∴ a = 2 or -2

 

Example 7: Solve: 4 × 3^x+1 = 27 + 9^x

Solution: Here,

                   4 × 3^x+1 = 27 + 9^x

          or,     9^x – 4 × 3^x+1 + 27 = 0

          or,     3^2x – 4 . 3^x . 3¹ + 27 = 0

          or,     (3^x)² – 12 . 3^x + 27 = 0

          Let, 3^x = 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, 3^x = 3²

          ∴ x = 2

Or, a – 3 = 0

          or, a = 3

          or, 3^x = 3¹

          ∴ x = 1

∴ x = 1 or 2

 

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

 

'Indices' is the plural form of 'index'. And, index means the power of any number or variable term, which is called a base. Power or index is also known as an exponent.

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10 Math Problems officially announces the release of Quick Math Solver and 10 Math Problems, Apps on Google Play Store for students around the world.

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