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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

User Validcat
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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)* 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)

User Alexey Kuznetsov
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