I need help solving a logarithm equation.

Question

these were my steps. Can someone tell me where have i goe wrong since the answer is $-{\frac 14}$.

$\sqrt{\log(\sqrt{10}a)} = {\frac 12}$

${\frac 12}{\log(\sqrt{10}a)} = \log\sqrt{10}$

${\log(\sqrt{10}a)}^{\frac 12} = {\log\sqrt10}$ because $(\log_ax^n = n\log_ax)$

$(\sqrt{10}a)^{\frac 12} = 10^{\frac 12}$ (squaring)

$10^{\frac 12}a = 10$ division by $10^{\frac 12}$

$ a = 10^{1-\frac 12}$

$a = \sqrt{10}$

Answer 1

$\sqrt{\log(\sqrt{10}a)}=\frac12$

$\log(\sqrt{10}a)=\frac14$

$\sqrt{10}a=10^{1/4}$

$a=10^{-1/4}$

Answer 2

The first step you take is incorrect.

\begin{align} \sqrt{\log(\sqrt{10}a)} & = {\frac 12} \\ \log(\sqrt{10}a) & = \frac{1}{4} \\ \log(\sqrt{10}a) & = \log(10^{\frac 14}) \\ \sqrt{10}a = 10^{\frac 14} \\ a = 10^{-\frac 14} \end{align}