• DeFi protocols often have math operations involving logarithm and exponent. Prominent examples are Uniswap and Balancer. They handle millions of dollars' TVL and trading volume.
• 3rd party libraries like ABDK were developed because Solidity lacks native log and exponent function.
• Existing libraries use a combination of iteration, bit shifting and 'magic numbers'. However they're complex gas intensive. Here's Uniswap's log function 🤯

    function getTickAtSqrtRatio(uint160 sqrtPriceX96) internal pure returns (int24 tick) {
        // second inequality must be < because the price can never reach the price at the max tick
        require(sqrtPriceX96 >= MIN_SQRT_RATIO && sqrtPriceX96 < MAX_SQRT_RATIO, 'R');
        uint256 ratio = uint256(sqrtPriceX96) << 32;

        uint256 r = ratio;
        uint256 msb = 0;

        assembly {
            let f := shl(7, gt(r, 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF))
            msb := or(msb, f)
            r := shr(f, r)
        }
        assembly {
            let f := shl(6, gt(r, 0xFFFFFFFFFFFFFFFF))
            msb := or(msb, f)
            r := shr(f, r)
        }
        assembly {
            let f := shl(5, gt(r, 0xFFFFFFFF))
            msb := or(msb, f)
            r := shr(f, r)
        }
        assembly {
            let f := shl(4, gt(r, 0xFFFF))
            msb := or(msb, f)
            r := shr(f, r)
        }
// lot more code

• Enter Napier's bones or log tables. Popular in elementary school math, you lookup values in the table instead of calculating.
• Napier's bones meets merkle tree. I built a merkle tree of the log10 table and stored its root on chain.
• Now we can look up log off-chain and pass it to the smart contract with a proof. Merkle tree ensures that the log is cryptographically valid.

The result is around 50% improvement in gas over ABDK, a significant improvement.