[Math] How to solve $\log \sqrt[3]{x} = \sqrt{\log x} ?$ algebra-precalculuslogarithms How to solve $$\log \sqrt[3]{x} = \sqrt{\log x} $$ Best Answer Using $$m\log a=\log(a^m)$$ when both logs are defined $$\log\sqrt[3] x=\sqrt{\log x}\implies\frac13 \log x=\sqrt{\log x}$$ $$\sqrt{\log x}(\sqrt{\log x}-3)=0$$ $$\sqrt{\log x}=0\iff \log x=0\iff x=1$$ $$\sqrt{\log x}-3=0\iff \log x=9$$ Related Solutions[Math] How to simplify $\log (1/\sqrt{1000})$ Hint: $$\frac{1}{\sqrt{1000}}=10^{-\frac{3}{2}}\qquad\mbox{and}\qquad\log x^a=a\log x$$ [Math] How to solve the equation $x \log \log x = n$ With the estimate $x=\dfrac n{\log(\log(n))}$, you have $$\dfrac n{\log(\log(n))}\log\left(\log\left(\dfrac n{\log(\log(n))}\right)\right)=\dfrac n{\log(\log(n))}\log\left(\log(n)-\log\left(\log(\log(n))\right)\right)$$ which is asymptotically $n$. For example, for $n=100$, $$x=\frac{100}{\log(\log(100))}=65.4801821\cdots$$ and $$x\ln(\ln(x))=93.684410\cdots.$$ Related QuestionSolve log equation $\frac{3}{log_2(10)}-log(x-9)=log(44)$
Best Answer
Using $$m\log a=\log(a^m)$$ when both logs are defined
$$\log\sqrt[3] x=\sqrt{\log x}\implies\frac13 \log x=\sqrt{\log x}$$
$$\sqrt{\log x}(\sqrt{\log x}-3)=0$$
$$\sqrt{\log x}=0\iff \log x=0\iff x=1$$
$$\sqrt{\log x}-3=0\iff \log x=9$$