Let's take the problem step by step.
(a) log_9(4) + log_9(7) = log_9(what?)
Using the rule of logarithms which states that the sum of two logs with the same base is equal to the log of the product of the numbers they hold, we rewrite the equation as:
log_9(4*7) = log_9(what?)
This simplifies to:
log_9(28) = log_9(what?)
Therefore, the unknown value in this equation for which it holds true is 28.
(b) log_8(what?) - log_8(5) = log_8(11/5)
Here we use another rule of logarithms which states that the difference of two logarithms is the log of the quotient of the numbers they hold. We can hence rewrite the equation as:
log_8((what?)/5) = log_8(11/5)
Solving for 'what?', we get the equation (what?)/5 = 8/5. Therefore, 'what?' = 8, and our missing value is replaced with 1.6.
(c) log_9(8) = 3 * log_9(what?)
Using a third logarithm property, we can write the equation as:
log_9(8) = log_9((what?)³)
Therefore (what?)³ = 8. Using the cubed root, we find that 'what?' = 2.
So the proper replacement for the values making the equations true are: (a) 28, (b) 1.6, and (c) 2.0.