JEE Advanced 2022 Paper 2 · Q04 · Logarithms

Question

The product of all positive real values of $x$ satisfying the equation $x^{\left(16(\log_{5}x)^{3}-68\log_{5}x\right)}=5^{-16}$ is _____.

SolveKar AI · Accepted Answer

Take $\log_5$ on both sides: let $t=\log_5 x$. Then $t\cdot(16t^3-68t)=-16$, i.e. $16t^4-68t^2+16=0$, i.e. $4t^4-17t^2+4=0$. This is quadratic in $t^2$: $t^2=\dfrac{17\pm\sqrt{289-64}}{8}=\dfrac{17\pm 15}{8}$, giving $t^2=4$ or $t^2=1/4$. So $t\in\{2,-2,1/2,-1/2\}$. The roots of $\log_5 x = t$ are $x=5^t$. The product of all four values is $5^{2+(-2)+1/2+(-1/2)}=5^0=1$.

JEE Advanced 2022 Paper 2 · Q04 · Logarithms

Solution