Exponential Equation Formula (original) (raw)
Last Updated : 10 Jan, 2024
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,
- 3x = 243
- 5x - 3 = 125
- 6y - 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 = 12 2
- 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 = 12 2
- 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 log 2 9 = x
**Sample Problems
**Question 1. Solve the exponential equation: 10 x = 10 10 .
**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.
⇒ 6z - 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 5x = log 4
As per the property log am = 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 73x + 7 = log 490
As per the property log am = 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