Równanie logarytmiczne

[Januszgolenia]

\(log_x{10}+2log_{10x}{10}-3log_{100x}{10}=0\)

[eresh]

\(log_x{10}+2log_{10x}{10}-3log_{100x}{10}=0\)

\(\log_x{10}+2\log_{10x}{10}-3\log_{100x}{10}=0\\
x> 0\;\;\wedge x\neq 1\;\; \wedge \;\;x\neq\frac{1}{10}\;\;\wedge\;\;x\neq \frac{1}{100}\\
\frac{1}{\log x}+\frac{2}{\log (10x)}-\frac{3}{\log (100x)}=0\\
\frac{1}{\log x}+\frac{2}{1+\log x}-\frac{3}{2+\log x}=0\\
\frac{(1+\log x)(2+\log x)+2\log x(2+\log x)-3\log x(1+\log x)}{\log x(1+\log x)(2+\log x)}=0\\
2+\log x+2\log x+\log^2x+4\log x+2\log^2x-3\log x-3\log^2x=0\\
4\log x+2=0\\
\log x=-\frac{1}{2}\\
x=10^{-\frac{1}{2}}\\
x=\frac{1}{\sqrt{10}}
\)

[Galen]

Można też wprowadzić zmienną pomocniczą...
\(u=\log_x10\;\;\;\;\;\;\;\;i\;\;\;\;\;\;\;\;\;u\neq 0\)
\(u+\frac{2u}{u+1}- \frac{3u}{2u+1}=0\;/:u\\1+ \frac{2}{u+1}- \frac{3}{2u+1}=0\;/\cdot(u+1)(2u+1)\\
...\\2u^2+4u=0\\2u(u+2)=0\\u=-2\\log_x10=-2\\x^{-2}=10\\ \frac{1}{x^2}= \frac{10}{1}\\10x^2=1\\x^2= \frac{1}{10}\;\;\;i\;\;\;x>0\\x= \frac{1}{ \sqrt{10} }\)