Final answer:
To compute the expression ⁴/a b ÷ ³/b a, substitute the values of a and b into the expression. Simplify using the properties of logarithms and exponentials, and the answer is 3.
Step-by-step explanation:
To compute ⁴/a b ÷ ³/b a, we need to substitute the values of a and b into the expression. Given that a = log₉ and b = log₁₆, we can rewrite the expression as ⁴/(log₉) (log₁₆) ÷ ³/(log₁₆) (log₉). Using the property of logarithms that the logarithm of the number resulting from the division of two numbers is the difference between the logarithms of the two numbers, we simplify the expression to ⁴(log₉ - log₁₆) ÷ ³(log₁₆ - log₉).
Next, we use the property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. We can rewrite the expression as ⁴(3(log₃ - log₂)) ÷ ³(3(log₂ - log₃)).
Simplifying further, we get ⁴(3log₃ - 3log₂) ÷ ³(3log₂ - 3log₃). Now, we can use the property that the cube of the exponentials involves cubing the digit term and multiplying the exponent by 3. Simplifying, we get 81(log₃ - log₂) ÷ 27(log₂ - log₃).
Finally, we can simplify further using the property that log() = log a - log b. Applying this property, we have 81(log₃ - log₂) ÷ 27(log₂ - log₃) = 81(log₃ - (log₃ - log₉)) ÷ 27((log₃ - log₉) - log₃). This simplifies to 81/27 = 3. Therefore, the answer is a) 3.