NIMCET 2025 — Mathematics PYQ

Step 1: Take Logarithm on Both Sides

Given the equation $x^{(8{\log }_5(x)-24)}=5^{-4}$, take ${\log }_5$ on both sides:

Using the property ${\log }_b(a^m)=m{\log }_b(a)$:

Step 2: Substitute and Form a Quadratic Equation

Let ${\log }_5(x)=t$. The equation becomes:

Divide the entire equation by $4$:

Step 3: Find the Sum of Roots

The roots of this quadratic equation are $t_1$ and $t_2$. These represent the possible values of ${\log }_5(x)$:

$t_1={\log }_5(x_1)$

$t_2={\log }_5(x_2)$

Using the sum of roots formula ($t_1+t_2=-b/a$):

Step 4: Find the Product of Values of $x$

Substitute the logarithmic forms back into the sum:

Using the property ${\log }_b(m)+{\log }_b(n)={\log }_b(mn)$:

Convert the logarithmic equation to exponential form:

Final Answer:

The product of all possible values of $x$ is $5^3$ (or $125$). (Option B)