Question

Certain functions obey the property f(m + n)=f(m)f(n) for all positive numbers and n.Can you think of a function that obeys this property? Hint: Functions that obey this property ill be familiar from ordinary pre-calculus algebra courses. Same question, but this time, the property is f(mn) f(m) +f(n) . (Note, don't expect f to be integer-valued. The hint from the first part applies here too.) m

Answer 1

Answer:

Given

$f(m+n)=f(m)f(n)$

If we assume

$f(x)=ae^{x}\\\\f(m+n)=ae^{m+n}\\\\\therefore f(m+n)=ae^{m}\times ae^{n}(\because x^{a+b}=x^{a}\times x^{b})\\\\\Rightarrow f(m+n)=f(m)\times f(n)$

Similarly

We can generalise the result for

$f(x)=am^{x }$ where a,m are real numbers

2)

$f(m\cdot n)=f(m)+f(n)\\\\let\\f(x)=log(x)\\\therefore f(mn)=log(mn)=log(m)+log(n)\\\\\therefore f(mn)=f(m)+f(n)$