## Task B-3.3.

Determine all real numbers $x$ for which $3^{\frac{1}{\log x}}-2 \cdot 3^{\frac{\log 10 x^{2}}{\log x^{2}}}=27$.
## Solution.

Apply the rules for logarithms

$3^{\frac{1}{\log x}}-2 \cdot 3^{\frac{\log 10+\log x^{2}}{\log x^{2}}}=27$

$3^{\frac{1}{\log x}}-2 \cdot 3^{\frac{1}{\log x^{2}}+1}=27$

$3^{\frac{1}{\log x}}-6 \cdot 3^{\frac{1}{\log x}}-27=0$

Introduce the substitution $3^{\frac{1}{2 \log x}}=t$ 1 point

The solutions to the quadratic equation $t^{2}-6 t-27=0$ are $t_{1}=-3$ and $t_{2}=9$ 1 point

For $t_{1}=-3$ there are no solutions.

For $t_{2}=9$ we get

$3^{\frac{1}{\log x}}=3^{2}$

$\frac{1}{2 \log x}=2, \log x \neq 0$

$\log x=\frac{1}{4}$.

Finally, we get $x=10^{\frac{1}{4}}=\sqrt[4]{10}$.