I'm trying to think of a card game based on using arithmetic together with the imaginary unit to compete against another player to make certain numbers using only other numbers and a set of operators. So, this game focuses heavily on using inverse operations and using powers of the imaginary unit to create negative numbers and inverse fractions.
For example, you can add two positive integers a and b as $$a + b$$ Otherwise, you can use subtraction with the additive identity $0$ to get $$a - (0 - b)$$ Using the imaginary unit, you can make b negative and get $$a - ((i^2)b)$$ With multiplication and division, you can multiply two positive integers a and b as $$ab$$ You can use the multiplicative identity 1 to get multiplication from division $$a \div (\frac{1}{b})$$ You can also use powers of $i^2$ or raising to $i$ twice to get the multiplicative inverse $$\frac{a}{(b^i)^i}$$ or $$\frac{a}{b^{i^2}}$$
So, my question, then, is about other uses of identities such as $0$ and $1$, together with other inverse processes like exponents and logarithms, or trigonometric functions and their inverses to do something similar. For example, with two positive integers, a and b, how can you go back and forth between exponents and logarithms using $i$ and basic identities like $0$ and $1$? Suppose you can only use logarithms, how do you take $a$, $b$, and the logarithm operation and end up with $a^b$? Or, conversely, suppose you can only use exponents, how do you take $a$, $b$, and the raise operation and end up with $\log_{a}{b}$?
