równanie z logarytmami

[maweave]

\(log(5^{ \sqrt{x}} + 1) - 2log5 = log(5^{1- \sqrt{x}} + 5)\)

[irena]

\(log(5^{\sqrt{x}}+1)-2log5=log(5^{1-\sqrt{x}}+5)\\log\frac{5^{\sqrt{x}}+1}{25}=log(\frac{5}{5^{\sqrt{x}}}+5)\\\frac{5^{\sqrt{x}}+1}{25}=\frac{5}{5^{\sqrt{x}}}+5\ /\cdot25\cdot5^{\sqrt{x}}\\5^{2\sqrt{x}}+5^{\sqrt{x}}=125+125\cdot5^{\sqrt{x}}\\5^{\sqrt{x}}=t>0\\t^2-124t-125=0\\(t+1)(t-125)=0\\t=-1<0\ \vee\ t=125\\t=125\\5^{\sqrt{x}}=125\\5^{\sqrt{x}}=5^3\\\sqrt{x}=3\\x=9\)