Question:
I am a mathematics teacher and one of my students recently
came back from a university interview with the following question he had
been asked orally. As far as I can tell he was rejected from a place on
the basis of his inability to answer the question. I think he got the
question wrong when he presented it to me (English is not his first
language) but as always in such cases I could be wrong. I know I cannot
figure out an answer and as far as I can tell there is none. Can you help?

Replies:
Find a non-constant function such that [f(x)]^a = af(x).

He was not completely clear whether or not "a" was a variable or a fixed
value. He seemed to think it was any old number.

Starting with [f(x)]^a = a*f(x). Take the 'log' of both sides gives:
log{[f(x)^a]} = log{a*f(x)}

Using the properties of the 'log' function that: log{x^a} = a* log{x} and
log{a*x} = log {a} +
log{x} gives:

a*log{f(x)} = log{a} + log{f(x)}

Subtracting log{f(x)} from both sides gives:

a*log{f(x)} - log{f(x)} = log{a} . Collecting terms on the left hand side
gives:

log{f(x)} * [a -1] = log{a} . Dividing by [a-1] providing a not = 1 gives:

This gives f(x) in terms of 'a' provided 'a' not = 1. However, since 'x'
does not appear on the right hand side, without knowing some additional
relation between 'x' and 'a' those variables have been separated and no
solution exists.

Vince Calder
I think the answer they were looking for is f(x) = ln(x).

Tim Mooney
If you take the logarithm of both sides you would get:
f[x]=Exp[ln(a)/(a-1)].
As such, there is no non-constant answer that satisfy this equation.

AK
Ali Khounsary, Ph.D.
Advanced Photon Source
Argonne National Laboratory
If we solve the equation for f(x) by first taking the logorithm of both
sides, we get
f(x) = exp{[ln(a)]/(a-1)}.
If a is a constant, then f(x) is a constant. This may be what they were
looking for.

Another possibility may be a search for possible values of a. In this
problem, the only constant value of a that will work for a non-constant f(x)
is a=1.

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