JEE Main Logarithms Question | Change of Base PYQ

 

❓ Question

If

$$\frac{\log_{8}\left(\frac{8}{x^{2}}\right)} {\left(\log_{8}x\right)^2}=3,$$

find the product of the roots.

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✍️ Solution

Using the logarithm property,

$$\log_b\left(\frac{a}{c}\right)=\log_ba-\log_bc,$$

the given equation becomes

$$\log_8 8-\log_8 x^2 = 3\left(\log_8x\right)^2.$$

Since

$$\log_88=1,$$

and

$$\log_8x^2=2\log_8x,$$

we get

$$1-2\log_8x = 3\left(\log_8x\right)^2.$$

Let

$$t=\log_8x.$$

Then,

$$1-2t=3t^2,$$

or

$$3t^2+2t-1=0.$$

Factorizing,

$$(3t-1)(t+1)=0.$$

Hence,

$$t=\frac13 \quad\text{or}\quad t=-1.$$

Therefore,

$$\log_8x=\frac13 \Rightarrow x=8^{1/3}=2,$$

and

$$\log_8x=-1 \Rightarrow x=8^{-1}=\frac18.$$

Hence, the product of the roots is

$$2\times\frac18=\frac14.$$

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✅ Final Answer

$$\boxed{\frac14}$$