Exponential Functions

Exponential Functions: Level 4 Challenges


What is the sum of all integer values of xx such that

(x217x+71)(x234x+240)=1? (x^2-17x+71)^{(x^2-34x+240)} =1 ?

Details and assumptions

xx can be a negative integer. Since the exponent is an integer, the value is still well defined.

000^0 is undefined.

a(a1)(a2)=aa23a+2\Large a^{(a-1)^{(a-2)}} = a^{a^2-3a+2}

Find the sum of all positive integers aa that satisfy the equation above.

 If xxx16=16, then evaluate xxx12. \Large \text{ If } x^{x^{x^{16}}} = 16, \text{ then evaluate } x^{x^{x^{12}}}.

Note: xx is a real number.

In England, with respect to the initial population each year, the death rate is 146\frac{1}{46} and the birth rate is 133.\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 am+n=am×ana^{m+n}=a^m \times a^n But there are some students who claim that am+n=am+ana^{m+n}=a^m+a^n which is obviously incorrect. However, it is true for some triplets of integers, (a,m,n)(a,m,n). If a[0,10]a\in[0,10] and m,n[1,10]m,n\in[1,10], then find the number of possible triplets for which am+n=am+ana^{m+n}=a^m+a^n.


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