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Let f be a function such that $f(mn) = f(m) f(n)$ for every positive integers m and n. If $f(1), f(2)$ and $f(3)$ are positive integers, $f(1) < f(2),$ and $f(24) = 54$, then $f(18)$ equals ?

Process:-

I attempted to solve this question using 2 approaches , but couldn't reach to the answer using approach 2

Approach 1:-

$f(24) = f(2^3 \cdot 3) = f(2^3) \cdot f(3) = [f(2)]^3 \cdot f(3) = 54$

Similarly writing $f(18)$ as $f (2 \cdot 3^2) = f(2) \cdot [f(3)]^2$

Also $f(1 \cdot 2) = f(1) \cdot f(2) > f(1)\Rightarrow f(1) \cdot [f(2) - 1] > 0,$ now as $f(1)$ is a positive integer we get $f (2) > 1$ , also $f(2) < 4$ , so we are left with $f(2) = 2$ or $f(2) = 3$

by using $f(2) = 2,$ we don't get $f(3) $ as an integer , and $f(2) = 3$ we get $f(3) = 2$

so finally putting $f(2)=3$ and $f(3)=2$ in the expression of $f(18)$ we get the value as $3 \cdot 4 = 12$

My 2nd approach :-

Using the result that $f(x y) = f(x) \cdot f(y)$ gives us a function of the form $f(x) = x ^ t$ , where $x, y$ are positive integers and t is a real number { I am not sure if I am using the condition correctly in this step , please correct me if wrong }

so $24 ^ t = 54$ $\Rightarrow t = \frac{\log (54)}{ \log (24) }$

and now we have to find $f(18)$ so it would be $18 ^ t = 18 ^{\frac{\log (54)}{ \log (24)}} = 37.63 $

where exactly am I going wrong in my 2nd approach , also please clarify when does $f(xy) = f(x) f(y)$ gives us a function of the form $x ^ t.$

Fin27
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1 Answers1

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With $f(2)=3$ and $f(3)=2$ the form of $f(x)=x^t$ is not the correct form.

Note that $2^t=3 \implies t=\log_2 3$ while $3^t=2 \implies t=\log_3 2$

We do not necessarily get a closed form for this function.

Mohammad Riazi-Kermani
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