Exponential functional equations?

Algebra Level 4

$\large f(2^x)=2^{f(x)}$

Let $f$ satisfy the equation above with $f(0) = 1$ . If $f(2048)$ can be expressed as $a^b$ , where $a$ and $b$ are positive integers with $a$ prime, find $a+b$ .

The answer is 2050.

1 solution

Akash Shukla
Jun 7, 2016

$f(2^{x}) = 2^{f(x)}$

$f(0) = 1$

$f(2^{0}) = 2^{f(0)} = 2^{1}$

Similarly, $f(2^{1}) = 2^{f(1)} = 2^{2}$

$f(2^{2}) = 2^{f(2)} = 2^{2^{2}}$

$f(2^{4}) = 2^{f(4)} = 2^{2^{2^{2}}} = 2^{2^{4}}$

Hence, $f(2048) = 2^{2048}$

$2+2048 = 2050$

Exponential functional equations?

The answer is 2050.

1 solution

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