The property of equality for exponential equations states that if two exponential expressions have the same base and are equal, then their exponents must also be equal.
In simpler terms: If bx = by, then x = y.
Explanation
This property is fundamental in solving exponential equations. It allows us to equate the exponents once the bases are the same, transforming the exponential equation into a simpler algebraic equation.
Formal Definition
For any real number b where b > 0 and b ≠ 1, if bx = by, then x = y.
Examples
-
Example 1: Solve for x in the equation 2x = 25.
- Since the bases are the same (both are 2), we can equate the exponents: x = 5.
- Therefore, the solution is x = 5.
-
Example 2: Solve for x in the equation 3x+1 = 37.
- Since the bases are the same (both are 3), we can equate the exponents: x + 1 = 7.
- Solving for x, we get x = 7 - 1 = 6.
- Therefore, the solution is x = 6.
-
Example 3: Solve for x in the equation 4x = 26.
- First, express both sides with the same base. Since 4 = 22, we can rewrite the equation as (22)x = 26.
- Using the power of a power rule, we get 22x = 26.
- Now, the bases are the same. Equate the exponents: 2x = 6.
- Solving for x, we get x = 6 / 2 = 3.
- Therefore, the solution is x = 3.
Conditions for the Property to Hold
The property of equality for exponential equations holds true under the following conditions:
- The bases must be the same.
- The base b must be a positive real number.
- The base b cannot be equal to 1. (Because 1 raised to any power is always 1).
Importance
This property is a critical tool in solving various mathematical problems, especially those involving exponential growth and decay, compound interest, and radioactive decay. It simplifies complex equations, making them easier to solve.
In conclusion, the property of equality of exponential equations is a powerful rule that allows us to solve exponential equations by equating the exponents when the bases are the same, enabling us to simplify and find solutions more easily.
