The product of all positive real values of $x$ satisfying the equation $x^{\left(16\left(\log _5 x\right)^3-68 \log _5 x\right)}=5^{-16}$ is _____.
Correct Answer: 1
Solution and Explanation
Taking log5 on both sides
\((16(\log_5x)^3 -68(\log_5x))(\log_5x) = -16\)
Let ( log5x)=t
16t4 - 68t2 + 16 = 0
4t4 + 16t2 - t2+4=0
(4t2 - 1)(t2-4) = 0
\(t=±\frac{1}{2}\ or ±2\)
So log5x = \(±\frac{1}{2}\ or ±2\)
\(⇒ x=5^{\frac{1}{2}} , 5^{-\frac{1}{2}} , 5^2, 5^{-2}\)
\(\therefore\) Product = \((5)^{\frac{1}{2}-\frac{1}{2} +2-2}\)
\(= 5^0\)
\( = 1\)
So, the correct answer is 1.
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Concepts Used:
Logarithmic Differentiation
Logarithmic differentiation is a method to find the derivatives of some complicated functions, using logarithms. There are cases in which differentiating the logarithm of a given function is simpler as compared to differentiating the function itself. By the proper usage of properties of logarithms and chain rule finding, the derivatives become easy. This concept is applicable to nearly all the non-zero functions which are differentiable in nature.
Therefore, in calculus, the differentiation of some complex functions is done by taking logarithms and then the logarithmic derivative is utilized to solve such a function.