If f(x+y)=2 f(x) f(y), f^'(5)=1024(log2) and f(2)=8, then the value of f(3) is (a) 64(log2) (b)

step 1 and f of x equals log base 2 of x, and I want to decide what f of 1 eighth equals. So I'm replac

If f(x+y)=2 f(x) f(y), f^'(5)=1024(log2) and f(2)=8, then...

Key Concepts

Functional Equations

Functional equations involve relationships that a function must satisfy for all inputs in its domain. In this context, an equation like f(x+y) = constant * f(x) * f(y) suggests the function may have an exponential nature since exponentials naturally convert sums into products.

Exponential Functions

Exponential functions are characterized by the property that the function’s value grows or decays at a rate proportional to its current value. Their form, often written as f(x) = A * e^(kx) or variations of this, enables them to satisfy multiplicative conditions derived from additive arguments, as seen in the equation f(x+y) = 2 f(x) f(y).

Differentiation and Derivatives

Differentiation is the mathematical process of finding the derivative of a function, which measures the rate at which the function changes with respect to its input. Given a specific derivative value at a certain point, this concept is essential for determining unknown parameters or constants in the function that meet both the functional equation and the derivative condition.

Logarithms

Logarithms are the inverse operations of exponentiation and are useful in solving exponential equations. When dealing with exponential functions, logarithms can help linearize multiplicative relationships, which is particularly crucial when matching derivative values that involve logarithmic expressions.

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