> [!summary] Indeterminate powers such as $0^0,\infty^0$ and $1^\infty$ can be solved using properties of logarithms to rewrite the limit into easier indeterminate forms, which have techniques to solve. >[!info]+ Read Time **⏱ 1 min** # Definition Indeterminate powers are [[Limits|limits]] that result in [[Indeterminate Forms|indeterminate forms]], $0^0,\infty^0$, and $1^\infty$. To solve these indeterminate forms, [[Logarithm Properties|properties of logarithms]] are used. This rewrites the limit into other indeterminate forms that techniques can be used on. Formally, if a limit had an indeterminate power such as $L = \displaystyle \lim_{ x \to a} \left[ f(x)\right]^{g(x)}$ then $ \begin{align*} L &= \displaystyle \lim_{ x \to a} \left[ f(x)\right]^{g(x)} \\ \ln(L) &= \displaystyle \lim_{ x \to a} \ln \left( \left[ f(x)\right]^{g(x)} \right) \\ \ln(L) &= \displaystyle \lim_{ x \to a} g(x) \ln \left( f(x) \right) \end{align*} $ The limit gets converted into other indeterminate forms (e.g. [[Indeterminate Products|indeterminate products]]). Which then using techniques like [[Multiplying by the Conjugate|multiplying by the conjugate]] or [[L'Hôpital's Rule|l'hospital's rule]] can be used. > [!note] After using techniques to solve the limit, you must solve for $L$ to give an proper result.