Solving logarithmic equation: sqrt[log(x) - 3] = log(x) - 3

[WTF?]

Hellos everyone, I'm in dire need of help, as I do not understand how to approach this. The problem is:

\(\displaystyle \sqrt{log(x)-3}\\) = \(\displaystyle log(x)-3\)

My first step was to square the square root of log(x) - 3 so that I would square the other side and get (log(x)-3)(log(x)-3)

Then, I'd cancel the logs to get (x)-3 = log(x)-3 + log(x) - 3

But i'm not sure how the book got 1000, or 10,000 :?

 

[royhaas]

Re: Simple logarithmic equations...

When you square both sides you get that \(\displaystyle \log(x)-3 = (log(x)-3)^2\). This is like \(\displaystyle y=y^2\) which has two solutions.

 

[soroban]

Re: Simple logarithmic equations...

Hello, WTF?!

\(\displaystyle \sqrt{\log(x)-3} \:=\: \log(x)-3\)

\(\displaystyle \text{We have: }\\log x - 3) - \sqrt{\log x - 3} \:=\:0\)


\(\displaystyle \text{Let: }\:u \:=\:\sqrt{\log x - 3} \quad\Rightarrow\quad u^2 \:=\:\log x - 3\)

\(\displaystyle \text{Substitute: }\;u^2 - u \:=\:0 \quad\Rightarrow\quad u(u-1) \:=\:0 \quad\Rightarrow\quad u \:=\:0,\:1\)


\(\displaystyle \text{Back-substitute:}\)

. . \(\displaystyle \log x - 3 \:=\:0 \quad\Rightarrow\quad \log x \:=\:3 \quad\Rightarrow\quad x \:=\:1,\!000\)

. . \(\displaystyle \log x - 3 \:=\:1 \quad\Rightarrow\quad \log x \:=\:4 \quad\Rightarrow\quad x \:=\:10,\!000\)

 

[WTF?]

Thank you, that helped a lot.