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.
Top Questions on Logarithmic Differentiation
- If $ f(x) = e^{2x} \sin x $, find $ f'(x) $.
- BITSAT - 2025
- Mathematics
- Logarithmic Differentiation
- If $ y = \ln(x^2 + 1) $, then find $ \frac{dy{dx} $ at $ x = 1 $.
- BITSAT - 2025
- Mathematics
- Logarithmic Differentiation
- Find the derivative of \[ \frac{d}{dx} \left( \log_3 x \cdot \log_x 3 \right). \]
- Bihar Board XII - 2025
- Mathematics
- Logarithmic Differentiation
- Find the derivative of \[ \frac{d}{dx} \left( \log x^{100} \right). \]
- Bihar Board XII - 2025
- Mathematics
- Logarithmic Differentiation
- Evaluate the derivative of $ y = \cos x \times \sin y, \quad \frac{dy}{dx} \text{ at } \left( \frac{\pi}{6}, \frac{\pi}{5} \right) $
- KEAM - 2025
- Mathematics
- Logarithmic Differentiation
Questions Asked in JEE Advanced exam
- Let $ x_0 $ be the real number such that $ e^{x_0} + x_0 = 0 $. For a given real number $ \alpha $, define $$ g(x) = \frac{3xe^x + 3x - \alpha e^x - \alpha x}{3(e^x + 1)} $$ for all real numbers $ x $. Then which one of the following statements is TRUE?
- JEE Advanced - 2025
- Fundamental Theorem of Calculus
- A linear octasaccharide (molar mass = 1024 g mol$^{-1}$) on complete hydrolysis produces three monosaccharides: ribose, 2-deoxyribose and glucose. The amount of 2-deoxyribose formed is 58.26 % (w/w) of the total amount of the monosaccharides produced in the hydrolyzed products. The number of ribose unit(s) present in one molecule of octasaccharide is _____.
Use: Molar mass (in g mol$^{-1}$): ribose = 150, 2-deoxyribose = 134, glucose = 180; Atomic mass (in amu): H = 1, O = 16- JEE Advanced - 2025
- Biomolecules
Let $ P(x_1, y_1) $ and $ Q(x_2, y_2) $ be two distinct points on the ellipse $$ \frac{x^2}{9} + \frac{y^2}{4} = 1 $$ such that $ y_1 > 0 $, and $ y_2 > 0 $. Let $ C $ denote the circle $ x^2 + y^2 = 9 $, and $ M $ be the point $ (3, 0) $. Suppose the line $ x = x_1 $ intersects $ C $ at $ R $, and the line $ x = x_2 $ intersects $ C $ at $ S $, such that the $ y $-coordinates of $ R $ and $ S $ are positive. Let $ \angle ROM = \frac{\pi}{6} $ and $ \angle SOM = \frac{\pi}{3} $, where $ O $ denotes the origin $ (0, 0) $. Let $ |XY| $ denote the length of the line segment $ XY $. Then which of the following statements is (are) TRUE?
- JEE Advanced - 2025
- Conic sections
- Adsorption of phenol from its aqueous solution on to fly ash obeys Freundlich isotherm. At a given temperature, from 10 mg g$^{-1}$ and 16 mg g$^{-1}$ aqueous phenol solutions, the concentrations of adsorbed phenol are measured to be 4 mg g$^{-1}$ and 10 mg g$^{-1}$, respectively. At this temperature, the concentration (in mg g$^{-1}$) of adsorbed phenol from 20 mg g$^{-1}$ aqueous solution of phenol will be ____. Use: $\log_{10} 2 = 0.3$
- JEE Advanced - 2025
- Adsorption
- At 300 K, an ideal dilute solution of a macromolecule exerts osmotic pressure that is expressed in terms of the height (h) of the solution (density = 1.00 g cm$^{-3}$) where h is equal to 2.00 cm. If the concentration of the dilute solution of the macromolecule is 2.00 g dm$^{-3}$, the molar mass of the macromolecule is calculated to be $X \times 10^{4}$ g mol$^{-1}$. The value of $X$ is ____. Use: Universal gas constant (R) = 8.3 J K$^{-1}$ mol$^{-1}$ and acceleration due to gravity (g) = 10 m s$^{-2}\}$
- JEE Advanced - 2025
- Colligative Properties
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.