Question

Plz solve this question​

Answer 1

Fining the value:

Step-by-step explanation:

$\ Given \ value:\\\\(x-2)^{x+1}=(x-2)^{x+3}\\\ Solution:\\\\(x-2)^{x+1}=(x-2)^{x+3}\\\\\ take \ log \ on \ both \ sides \\\\\rightarrow \log (x-2)^{x+1}= \log(x-2)^{x+3}\\\\\rightarrow (x+1) \log (x-2)= (x+3)\log(x-2)\\\\\rightarrow x\log (x-2)+\log (x-2) = x\log (x-2)+3\log(x-2)\\\\\rightarrow \log (x-2) = 3\log(x-2)\\\\\rightarrow \log (x-2) = \log(x-2)^3\\\\ \ take \ anti \log \\\\(x-2)=(x-2)^3\\\\(x-2) = t\\\\t= t^3\\t^3-t=0\\\\t(t^2-1) =0\\\\t=0 \ \ \ \ \ \ and \ \ \ \ \ t^2-1=0$

$t=0 \ \ \ \ \ \ and \ \ \ \ \ t^2-1=0\\\\x-2=0 \ \ \ \ \ and \ \ \ \ \ \ (x-2)^2 -1 =0 \\\\ x= 2 \ \ \ \ \ and \ \ \ \ \ x-2 = 1 \\\\ x= 2 \ \ \ \ \ and \ \ \ \ \ x = 3$

x ∈ 2 to ∞

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