Logarithm Properties PYQ | JEE Main Mathematics 2025

 

❓ Question

Find the value of

$$81^{\frac{1}{\log_5 3}} +27^{\log_9 36} +3^{\frac{4}{\log_7 9}}.$$

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✍️ Solution

First term

$$81^{\frac{1}{\log_5 3}} =(3^4)^{\frac{1}{\log_5 3}} =3^{\frac{4}{\log_5 3}}.$$

Using

$$\frac{1}{\log_b a}=\log_a b,$$

we get

$$3^{4\log_3 5}.$$

Using

$$a^{k\log_a b}=b^k,$$
$$3^{4\log_3 5}=5^4=625.$$

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Second term

$$27^{\log_9 36} =(3^3)^{\log_9 36} =3^{3\log_9 36}.$$

Now,

$$36=6^2,\qquad 9=3^2.$$

Hence,

$$3\log_9 36 = 3\cdot\frac{\log_3 36}{\log_3 9} = 3\cdot\frac{2\log_3 6}{2} = 3\log_3 6.$$

Therefore,

$$3^{3\log_3 6}=6^3=216.$$

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Third term

$$3^{\frac{4}{\log_7 9}} = 3^{4\log_9 7}.$$

Since

$$9=3^2,$$
$$\log_9 7=\frac{\log_3 7}{2}.$$

Thus,

$$3^{4\log_9 7} = 3^{2\log_3 7} = 7^2 = 49.$$

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Adding all three terms,

$$625+216+49=890.$$

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✅ Final Answer

$$\boxed{890}$$