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Exponential Functions

From compound interest to bubonic plague, things that grow or spread really fast are often modeled by exponential functions. Learn about these powerful functions (pun intended?).

Level 4

         

What is the sum of all integer values of \(x\) such that

\[ (x^2-17x+71)^{(x^2-34x+240)} =1 ? \]

Details and assumptions

\(x\) can be a negative integer. Since the exponent is an integer, the value is still well defined.

\(0^0\) is undefined.

\[\Large a^{(a-1)^{(a-2)}} = a^{a^2-3a+2}\]

Find the sum of all positive integers \(a\) that satisfy the equation above.

\[ \Large \text{ If } x^{x^{x^{16}}} = 16, \text{ then evaluate } x^{x^{x^{12}}}. \]

Note: \(x\) is a real number.

In England, with respect to the initial population each year, the death rate is \(\frac{1}{46}\) and the birth rate is \(\frac{1}{33}.\)

If there were no emigration, how many years would it take for the population to double?

Note: Round up your answer.

We know \[a^{m+n}=a^m \times a^n\] But there are some students who claim that \[a^{m+n}=a^m+a^n\] which is obviously incorrect. However, it is true for some triplets of integers, \((a,m,n)\). If \(a\in[0,10]\) and \(m,n\in[1,10]\), then find the number of possible triplets for which \(a^{m+n}=a^m+a^n\).

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