Then, we use the fact that exponential functions are one-to-one to set the exponents equal to one another, and solve for the unknown. Globally, the exponential map is not necessarily surjective. o = What is the rule of exponential function? exponential lies in $G$: $$ It seems $[v_1, v_2]$ 'measures' the difference between $\exp_{q}(v_1)\exp_{q}(v_2)$ and $\exp_{q}(v_1+v_2)$ to the first order, so I guess it plays a role similar to one that first order derivative $/1!$ plays in function's expansion into power series. How many laws are there in exponential function? Let's calculate the tangent space of $G$ at the identity matrix $I$, $T_I G$: $$ If you break down the problem, the function is easier to see: When you have multiple factors inside parentheses raised to a power, you raise every single term to that power. 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g The law implies that if the exponents with same bases are multiplied, then exponents are added together.