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  4. If log 2 √2(32 √[5]4)=α+β where α is an integer and β ∈[0,1), then α is

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Q. If $\log _{2 \sqrt{2}}(32 \sqrt[5]{4})=\alpha+\beta$ where $\alpha$ is an integer and $\beta \in[0,1)$, then $\alpha$ is

Continuity and Differentiability

Solution:

Given, $\log _{2^{\frac{3}{2}}}\left(2^5 \cdot 2^{\frac{2}{5}}\right)=\alpha+\beta=\log _{2^{\frac{3}{2}}}\left(2^{\frac{27}{5}}\right)=\frac{27 / 5}{3 / 2}=\frac{27}{5} \times \frac{2}{3}=\frac{18}{5}=3.6$
$\therefore \alpha=3, \beta=0.6$