Consider [tex]$8^{x-4}=8^{10}$[/tex]. Because the $\square$ are equal, the $\square$ must also be equal.
A. bases
B. exponents
C. equations
D. variables
Answer (2)
The bases are equal, so the exponents must be equal. Therefore, x − 4 = 10 .
Explanation
Understanding the Problem We are given the equation 8 x − 4 = 8 10 . We need to determine what must also be equal when the bases are equal.
Equating the Exponents Since the bases are equal, the exponents must be equal. This is a fundamental property of exponential functions: if a m = a n , then m = n , provided that a is not 0, 1, or -1. In our case, a = 8 , so we can equate the exponents.
Stating the Equality Therefore, x − 4 = 10 .
Examples
Exponential equations are used in various real-world scenarios, such as modeling population growth, radioactive decay, and compound interest. For example, if you invest money in a bank account with compound interest, the amount of money you have after a certain time can be modeled using an exponential equation. Understanding how to solve these equations allows you to predict future values and make informed decisions.
When the bases in the equation 8 x − 4 = 8 10 are equal, we can equate the exponents, leading to x − 4 = 10 . Solving this gives us x = 14 . Therefore, the answer to fill in the blanks is that the exponents must be equal.
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