Question
Solve for $x : \log (x + 5) + \log (x - 5) = 4 \log 2 + 2 \log 3$
✓
Answer
$\log (x+5)+\log (x-5)=4 \log 2+2 \log 3$
$\Rightarrow \log (x+5)(x-5)=4 \log 2+2 \log 3 \ldots\left[\log _a m+\log _a n+\log _a m n\right]$
$\Rightarrow \log \left(x^2-25\right)=\log 2^4+\log 3^2 \ldots\left[n \log _a m=\log _a m^n\right]$
$ \Rightarrow \log \left(x^2-25\right)=\log 16+\log 9$
$ \Rightarrow \log \left(x^2-25\right)=\log 16 \times 9 \ldots\left[\log _a m+\log _a n+\log _a m n\right]$
$ \Rightarrow \log \left(x^2-25\right)=\log 144$
$ \Rightarrow x^2-25=144$
$ \Rightarrow x^2=144+25$
$ \Rightarrow x^2=169$
$ \Rightarrow x= \pm \sqrt{169}$
$ \Rightarrow x= \pm \sqrt{13^2}$
$ \Rightarrow x= \pm 13$
$\Rightarrow \log (x+5)(x-5)=4 \log 2+2 \log 3 \ldots\left[\log _a m+\log _a n+\log _a m n\right]$
$\Rightarrow \log \left(x^2-25\right)=\log 2^4+\log 3^2 \ldots\left[n \log _a m=\log _a m^n\right]$
$ \Rightarrow \log \left(x^2-25\right)=\log 16+\log 9$
$ \Rightarrow \log \left(x^2-25\right)=\log 16 \times 9 \ldots\left[\log _a m+\log _a n+\log _a m n\right]$
$ \Rightarrow \log \left(x^2-25\right)=\log 144$
$ \Rightarrow x^2-25=144$
$ \Rightarrow x^2=144+25$
$ \Rightarrow x^2=169$
$ \Rightarrow x= \pm \sqrt{169}$
$ \Rightarrow x= \pm \sqrt{13^2}$
$ \Rightarrow x= \pm 13$
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