A function in which addition and multiplication behave the same way Exponents have a well-known property:
$$x^ax^b = x^{a+b}$$
but
$$x^{a} + x^{b} \neq x^{a+b}$$
Similarly,
$$\log(a) + \log(b) = \log(ab) $$
But
$$\log(a)\log(b) \neq \log(ab)$$
So my question is this:
Is there a function $f$ on $\mathbb{R}$ or some infinite subset of $\mathbb{R}$ with the following properties
$$(1)\quad f(x)f(y) = f(x+y)$$
$$(2)\quad f(x)+f(y) = f(x+y)$$
ie
$$(3)\quad f(x)+f(y) = f(x)f(y)$$
It seems that $(2)$ requires the function to be linear...

Any functions where both sum and product = 2 will work. zero will not work because it results in division by zero. $(x+y)/(x*y)-1=0$
