# How do you solve for x in 2(logx-log6)=logx-2logsqrt( x - 1)?

Feb 3, 2016

$x = \frac{1 + \sqrt{145}}{2} \approx 6.52$
(at least that's what I got)

#### Explanation:

Given
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{2 \left(\log \left(x\right) - \log \left(6\right)\right)} = \textcolor{b l u e}{\log \left(x\right) - 2 \log \left(\sqrt{x - 1}\right)}$

Thinks that are helpful to know:
[1]$\textcolor{w h i t e}{\text{XXX}} k \cdot \log \left(a\right) = \log \left({a}^{k}\right)$
[2]$\textcolor{w h i t e}{\text{XXX}} \log \left(a\right) - \log \left(b\right) = \log \left(\frac{a}{b}\right)$
[3]$\textcolor{w h i t e}{\text{XXX}} \log \left(\sqrt{a}\right) = \log \frac{a}{2}$ (this actually follows from [1])
[4]$\textcolor{w h i t e}{\text{XXX}}$for $\log \left(a\right)$ to be meaningful $a > 0$.

$\textcolor{red}{2 \left(\log \left(x\right) - \log \left(6\right)\right)}$

color(white)("XXX")=color(red)(2(log(x/6)) from [2]

color(white)("XXX")=color(red)(log((x^2)/(6^2)) from [1]

$\textcolor{b l u e}{\log \left(x\right) - 2 \log \left(\sqrt{x - 1}\right)}$

$\textcolor{w h i t e}{\text{XXX}} = \textcolor{b l u e}{\log \left(x\right) - 2 \left(\log \frac{x - 1}{2}\right)}$ from [3]

$\textcolor{w h i t e}{\text{XXX}} = \textcolor{b l u e}{\log \left(x\right) - \log \left(x - 1\right)}$ simplification

color(white)("XXX")=color(blue)(log(x/(x-1)) from [2]

Therefore
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{\log \left({x}^{2} / \left({6}^{2}\right)\right)} = \textcolor{b l u e}{\log \left(\frac{x}{x - 1}\right)}$

$\Rightarrow$$\textcolor{w h i t e}{\text{XXX}} {x}^{2} / 36 = \frac{x}{x - 1}$

$\Rightarrow$$\textcolor{w h i t e}{\text{XXX}} {x}^{3} - {x}^{2} = 36 x$

$\Rightarrow$$\textcolor{w h i t e}{\text{XXX}} x \left({x}^{2} - x - 36\right) = 0$

$\Rightarrow$$\textcolor{w h i t e}{\text{XXX")x=0color(white)("XX")orcolor(white)("XX}} \left({x}^{2} - x - 36\right) = 0$

But $x \ne 0$ from [4]
So
$\textcolor{w h i t e}{\text{XXX}} \left({x}^{2} - x - 36\right) = 0$

This can be factored using the quadratic formula to get
$\textcolor{w h i t e}{\text{XXX")x=(1+sqrt(145))/2color(white)("XX")orcolor(white)("XX}} x = \frac{1 - \sqrt{145}}{2}$

But $\frac{1 - \sqrt{145}}{2} < 0$ so $x \ne \frac{1 - \sqrt{145}}{2}$ from [4]

Therefore
$\textcolor{w h i t e}{\text{XXX}} x = \left(1 + \frac{\sqrt{145}}{2}\right) \approx 6.52$