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Q.
If $x+\frac{1}{x}=3$, then value of $\log _2\left(x^3+\frac{1}{x^3}+2\right)-\log _2\left(x^2+\frac{1}{x^2}+3\right)$ equals
Continuity and Differentiability
Solution:
$ x ^3+\frac{1}{ x ^3}+3(3)=27$
$\text { similarly, } x ^2+\frac{1}{ x ^2}=7 $
$x ^3+\frac{1}{ x ^3}=18 \Rightarrow \log _2\left( x ^3+\frac{1}{ x ^3}+2\right)-\log _2\left( x ^2+\frac{1}{ x ^2}+3\right)=\log _2 2=1$