Question

Find the value of x if ${\left(x^{{\log }_{10}3}\right)}^2-\left(3^{{\log }_{10}x}\right)-6=0$

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Answer 1

${\left(x^{{\log }_{10}3}\right)}^2-\left(3^{{\log }_{10}x}\right)-6=0$ ------------(1)

Let $x^{{\log }_{10}}3$ = t -------------(2)

Taking ${\log }_{10}$ on both sides

${\log }_{10}{x}^{{\log }_{10}3}$ = ${\log }_{10}t$

${\log }_{10}3$ $\times$ ${\log }_{10}x$ = ${\log }_{10}t$ ------------(3)

${\log }_{10}x$ $\times$ ${\log }_{10}3$ = ${\log }_{10}t$

${\log }_{10}{3}^{{\log }_{10}x}$ = ${\log }_{10}t$

We see that $3^{{\log }_{10}x}$ = t

So, from equation 1

$t^2$ - t - 6 = 0 --------------(4)

So t= 3 or -2
t=-2 is not possible.
putting t=3 in eqn 2 we get x=10 ( by taking log on both sides)