\(10^G=(10^{\lg \frac{U_2}{U_1}})^{20}\)

\(U_1^{20}=10^{-G}\cdot U_2^{20}\)

\(U_1=10^{-\frac{G}{20}}\cdot U_2\)

\(x^{\ln x}=e\)

\(\ln (x^{\ln x})=\ln e\)

\(\ln x \cdot \ln x=1\)

\(\ln x=\pm 1\)

\(x_1=e, x_2=\frac{1}{e}\)

\(\ln \left(\frac{2}{3}\ln x\right)=0\)

\(e^{\ln \left(\frac{2}{3}\ln x\right)}=e^0\)

\(\frac{2}{3}\ln x=1\)

\(\ln x=\frac{3}{2}\)

\(x=e^{\frac{3}{2}}\)

\(x^{\lg x}=\frac{x^3}{100}\)

\(100\cdot x^{\lg x}=x^3\)

\(\lg(100\cdot x^{\lg x})=\lg x^3\)

\(\lg100 + \lg x^{\lg x}=3\lg x\)

\(2 + \lg x\cdot \lg x-3\lg x=0\)

\(u^2-3u+2=0,\lg x=u\)

\(u_{1,2}=\frac{3}{2}\pm\sqrt{\frac{9}{4}-\frac{8}{4}}=\frac{3}{2}\pm\frac{1}{2}\)

\(\lg x_{1,2}=\frac{3}{2}\pm\frac{1}{2}\)

\(x_1=100,x_2=10\)

\(\ln (\ln x)=1\)

\(e^{e^{\ln (\ln x)}}=e^{e^1}\)

\(x=e^{e}\)

\(\lg^2 x-\lg x^5+6=0\)

\(\lg^2 x-5\lg x+6=0\)

\(u^2-5u+6=0,\lg x=u\)

\(u_{1,2}=2,5\pm\sqrt{6,25-6}=2,5\pm 0,5\)

\(\lg x=2,5\pm 0,5\)

\(x_1=10^3,x_2=10^2\)

\( 2^{log_2(x-5)}=2^7\\ x-5=128\\ x=128+5\\ x=133\)

\(p=\frac{1}{2}\ln\frac{P_1}{P_2}\)

\(e^p=e^{\ln\frac{P_1}{P_2}\frac{1}{2}}\)

\(e^p=(e^{\ln\frac{P_1}{P_2}})^{\frac{1}{2}}\)

\(e^p=(\frac{P_1}{P_2})^{\frac{1}{2}}\)

\(e^p=\frac{P_1^{\frac{1}{2}}}{P_2^{\frac{1}{2}}}\)

\(P_2^{\frac{1}{2}}=P_1^{\frac{1}{2}}\cdot e^{-p}\)

\(P_2=P_1\cdot e^{-2p}\)