we are given with the logarithmic equation
[tex] log4x+log4(x+6)=2 [/tex]
Using logarithmic Property
[tex] Loga+Logb=Logab [/tex]
we can write
[tex] Log[4x*4(x+6)]=2\\
\\
Log[16(x^2+6x)]=2\\
\\
[/tex]
Take anti-logarithm we get
[tex] 16(x^2+6x)=10^2\\
\\
16x^2+96x=100\\
\\
4x^2+24x=25\\
\\
4x^2+24x-25=0\\
\\ [/tex]
Now using Quadratic formula , we can find x as below:
[tex] x=\frac{-24 \pm\sqrt{24^2-4*4*(-25)}}{2*(4)}\\
\\
x= \frac{1}{8}(-24\pm4\sqrt{61}) [/tex]
As we know the logarithm is defined for the negative values.
Hence the possible solution is
[tex] x= \frac{1}{8}(-24+4\sqrt{61})\\
\\
x=0.905\\ [/tex]
