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Question

(C) Suppose,if possible,$\log_{2} 7$ is rational,say $p/q$ where $p$ and $q$ are integers,prime to each other.Then,$\frac{p}{q} = \log_{2} 7 \implies

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An irrational number

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(C) Suppose,if possible,$\log_{2} 7$ is rational,say $p/q$ where $p$ and $q$ are integers,prime to each other.Then,$\frac{p}{q} = \log_{2} 7 \implies 7 = 2^{p/q} \implies 2^{p} = 7^{q}$.This is a contradiction because the $L.H.S$ is an even number (power of $2$) and the $R.H.S$ is an odd number (power of $7$).Since it cannot be expressed as a ratio of two integers,$\log_{2} 7$ is an irrational number.Obviously,$\log_{2} 7$ is not an integer and hence not a prime number.

The number $\log_{2} 7$ is

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