JEE Main Logarithms Question | Solve in Under 2 Minutes

 

❓ Question

Evaluate

$$81^{\frac{1}{\log_3 5}} + 27^{\frac{\log 36}{\log 9}} + 3^{\frac{4}{\log_7 9}}.$$

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✍️ Solution

First term

Using

$$\frac{1}{\log_3 5}=\log_5 3,$$
$$81^{\frac{1}{\log_3 5}} =(3^4)^{\log_5 3} =3^{4\log_5 3}.$$

Now,

$$a^{\log_b c}=c^{\log_b a},$$

so,

$$3^{4\log_5 3} =3^{\log_5 81} =81^{\log_5 3} =5^{\log_5 81} =5^4=625.$$

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Second term

$$27^{\frac{\log36}{\log9}} =27^{\log_9 36}.$$

Since

$$36=6^2,\qquad 9=3^2,$$
$$\log_9 36=\log_3 6.$$

Hence,

$$27^{\log_3 6} =(3^3)^{\log_3 6} =6^3=216.$$

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Third term

$$3^{\frac{4}{\log_7 9}}$$

Using

$$\frac{1}{\log_7 9}=\log_9 7,$$
$$3^{4\log_9 7}.$$

Since

$$\log_9 7=\frac12\log_3 7,$$
$$=3^{2\log_3 7} =7^2=49.$$

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Adding all three terms,

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

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✅ Final Answer

$$\boxed{890}$$