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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?

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 $$a^{m+n}=a^m+a^n$$.