Final answer:
The exponential equations requested are already presented in the question. However, only option a) 4³ = 64 is correct, as the other options incorrectly claim to express the number 64 with given bases and exponents. In general, exponential and logarithmic forms are related inversely through their base, exponent, and result.
Step-by-step explanation:
The question revolves around converting logarithmic equations into exponential equations, which is a standard topic in high school math, particularly in algebra and pre-calculus. When we talk about logarithms and exponentials, we are dealing with two operations that are inverse to each other. The key relation is that if a = b^c, then log_b(a) = c. This means that logarithms give us the exponent when we know the base and the power result. Moreover, the equation of the form b^c = a is already an exponential equation, which is what is being asked for in the question.
Given the logarithmic equation log_4(64) = x, we want to express this in its exponential form. We use the rule that states if log_b(a) = x, then by definition b^x = a. Thus, for the example mentioned, the exponential form is 4^x = 64. In the cases provided, there is no need to convert as they are already in exponential form. For example, we directly see that 4 raised to the third power equals 64, which in exponential form is written as 4³ = 64.
The statement a) 4³ = 64 already represents an exponential equation where the base 4 is raised to the exponent 3 to equal 64. The statement b) 4¹/³ = 64 represents an exponential equation in which 4 is raised to the exponent 1/3, although this is incorrect since 4 raised to the 1/3 should equal 8. Statement c) repeats b) and hence has the same issue. Statement d) 2²/³ = 64 also represents an exponential equation, but it is incorrect since 2 raised to the 2/3 should equal 8. Hence, statements b), c), and d) appear to be incorrect or are wrongly stated if the intent is to express 64 in the form of an exponential equation with these bases and exponents.