Solve ( ln[sqrt(x + 4) + 2 ) / ln[ sqrt(x) ] = 2

[12345]

I am having a lot of trouble with this one math problem and hopefully someone here can help. Normally I am very good with logs, but this one I cannot get for some reason. Anyway:

. . .Solve ( ln[sqrt(x + 4) + 2 ) / ln[ sqrt(x) ] = 2

Thanks in advance for the help!

 

[soroban]

Re: logarithm help

Hello, 12345!

\(\displaystyle \frac{\ln\left(\sqrt{x+4}\,+\,2)}{\ln(\sqrt{x})}\;=\;2\)

First, note that: \(\displaystyle \,x\:\neq\:0\)

We have: \(\displaystyle \:\ln\left(\sqrt{x+4}\,+\,2\right) \:=\:2\cdot\ln(\sqrt{x}) \;=\;\ln(\sqrt{x})^2 \;=\;\ln(x)\)

Then: \(\displaystyle \:\sqrt{x+4}\,+\,2 \;=\;x\;\;\Rightarrow\;\;\sqrt{x+4} \;=\;x\,-\,2\)

Square both sides: \(\displaystyle \:x\,+\,4\;=\;x^2\,-\,4x\,+\,4\;\;\Rightarrow\;\;x^2\,-\,5x\;=\;0\)

Factor and solve: \(\displaystyle x(x\,-\,5)\:=\:0\;\;\Rightarrow\;\;x\:=\:0,\,5\)


Since \(\displaystyle x\,\neq\,0\), the only solution is: \(\displaystyle \:\fbox{x\,=\,5}\)

 

[tkhunny]

Or you could first note that x > 0 and x <> 1.