Solving $\log_x(a) = b $ and $\log_a(b) = x$ for $x$

Question

Solving for x:

$\log_a(x)=b \implies x=a^b$

$\log_x(a) = b \implies x = ?$

$\log_a(b) = x \implies x = ?$

Answer 1

The second gives $$x=a^{\frac{1}{b}}$$ because there is the following property. $$\log_{a^{\alpha}}b=\frac{1}{\alpha}\log_ab,$$ where $\alpha\neq0,$ $a>0$, $a\neq1$ and $b>0$.

Because for $x=a^{\frac{1}{b}}$ we obtain:$$\log_xa=\log_{a^{\frac{1}{b}}}a=\frac{1}{\frac{1}{b}}\log_aa=b.$$

The third gives $$x=\log_ab$$