Układ równań logarytmiczno-wykładniczo-potęgowy

[Januszgolenia]

\(( \sqrt{2})^x \cdot ( \sqrt[5]{5} )^y=80\text{ i } \log_ \frac{1}{3}(x-y)=-1\)

[eresh]

\(( \sqrt{2})^x \cdot ( \sqrt[5]{5} )^y=80 i log_ \frac{1}{3}(x-y)=-1\)

\(x-y>0\\x>y\)

\(\log_{\frac{1}{3}}(x-y)=-1\\
x-y=3\\
x=3+y\)

\((\sqrt{2})^x\cdot\sqrt[5]{5}^y=80\\
\sqrt{2}^{3+y}\cdot\sqrt[5]{5}^y=80\\
\sqrt{2}^3\cdot \sqrt{2}^y\cdot\sqrt[5]{5}^y=80\\
(\sqrt{2}\cdot\sqrt[5]{5})^y=20\sqrt{2}\\
(\sqrt{2}\cdot\sqrt[5]{5})^y=2^{\frac{5}{2}}\cdot 5\\
y=\log_{\sqrt{2}\cdot\sqrt[5]{5}}(2^{\frac{5}{2}}\cdot 5)\\
y=\log_{\sqrt{2}\sqrt[5]{5}}(\sqrt{2}\cdot\sqrt[5]{5})^5\\
y=5\\
x=3+5=8
\)