Which property of exponents states that a^x = a^y implies x = y?

The property of exponents known as the one-to-one property is vital for understanding how exponential equations function. This property asserts that if two exponential expressions with the same base are equal, then their exponents must also be equal. In mathematical terms, if ( a^x = a^y ), it follows that ( x ) must equal ( y ), provided ( a ) is a positive number and not equal to 1.

This property reflects that the exponential function is injective, or one-to-one; it does not allow for different exponent values to produce the same result when raised to the same base. This is essential in various applications, such as solving exponential equations, where one needs to deduce the value of the exponents from the equality of the exponential expressions.

The other properties mentioned—such as the power of a power property, the product of powers property, and the zero exponent property—do not relate to the one-to-one nature of equal bases leading to equal exponents. Instead, these properties describe how to manipulate powers in arithmetic, such as multiplying powers or raising powers to a power. Therefore, the one-to-one property is the correct explanation for the stated relationship between equal exponential expressions.