Exponential Equation Formula

Exponents are used in exponential equations, as the name implies. The exponent of a number (base) indicates how many times the number (base) has been multiplied. An exponential equation is one in which the power is a variable and is a part of an equation. 

Exponential Equations

A variable is the exponent (or a part of the exponent) in an exponential equation. For example, 

• 3^x = 243
• 5^{x – 3} = 125
• 6^{y – 7} = 216

The above examples depict exponential equations. Note how the variables x and y either form the entire exponent in the equation or just a part of it. Exponential equations are most commonly used to solve problems relating to compound interest, exponential growth, decay, etc.

Types of Exponential Equations

Exponential equations are classified into three categories. These are their names:

• Equations on both sides have the same base. These types of equations can be solved by equating their exponents. Example:

12^x = 12²

• Equations with distinct bases could be modified to have the same solution. Then when the bases have been equated, their exponents can be equated to solve for the variable. Example:

12^x = 144 can be represented as 12^x = 12²

• Equations that cannot be constructed to have the same base. These equations can be solved by applying a logarithm on both sides. Example:

2^x = 9 can be solved as log₂9 = x

Sample Problems

Question 1. Solve the exponential equation: 10^x = 10¹⁰.

Solution:

Clearly, the bases on both sides of the given equation are equal, then their exponents must also be equal.

Thus, x = 10.

Question 2. Solve: 6^{z – 7} = 216.

Solution:

We know that 216 = 6³.

⇒ 6^{z – 7} = 6³

Clearly, the bases on both sides of the given equation are equal, then their exponents must also be equal. 

⇒ z − 7 = 3

⇒ z = 3 + 7

⇒ z = 10

Question 3. Solve: (−5)^x = 625.

Solution:

We know: 625 = 5⁴ = (−5)⁴

⇒ (−5)^x = (−5)⁴

Clearly, the bases on both sides of the given equation are equal, then their exponents must also be equal. 

⇒ x = 4

Question 4. Solve: 5^x = 4.

Solution:

Since the bases cannot be made equal to each other in the given equation, we need to apply logarithms in order to solve for x.

⇒ log 5^x = log 4

As per the property log a^m = m log a, we have:

⇒ x log 5 = log 4

Divide both LHS and RHS by log 5.

⇒ x = log 4/log 5.

Question 5. Solve: 7^{3x + 7} = 490.

Solution:

Apply log on both sides of the given equation,

log 7^{3x + 7} = log 490

As per the property  log a^m = m log a, we have:

(3x + 7) log 7 = log 490 … (1)

x = -5/3 + (1/(3 log 7))

Question 6. Solve: 5^{x – 4} = 125.

Solution:

We know: 125 = 5³

⇒ (5)^x-4 = (5)³

Clearly, the bases on both sides of the given equation are equal, then their exponents must also be equal.

⇒ x − 4 = 3

⇒ x = 7

Question 7. Solve: 9^{n + 1} = 729.

Solution:

We know: 729 = 9³

⇒ (9)ⁿ⁺¹ = (9)³

Clearly, the bases on both sides of the given equation are equal, then their exponents must also be equal.

⇒ n + 1 = 3

⇒ n = 2