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

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)* ae^(n)(\because x^(a+b)=x^(a)* x^(b))\\\\\Rightarrow f(m+n)=f(m)* 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)`