Final answer:
The blank in the equation log_b(______) = d • log_b(a) can be filled with a^d, based on the logarithm property that relates logarithms with exponents. Hence, the correct answer is option C: a^d.
Step-by-step explanation:
The question is asking to fill in the blank with the correct expression that would make the equation logb(______) = d • logb(a) true for any positive numbers a, b (where b ≠ 1), and d. To find the correct expression, we need to use the properties of logarithms and exponents.
The third property of logarithms is particularly relevant here: The logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. Mathematically, this is expressed as logb(ax) = x • logb(a). Comparing this with the equation we have, we can deduce that the blank should be filled with ad, making the correct equation logb(ad) = d • logb(a), and therefore, the answer is option C: ad.