Logarithms Practice Test 1

Concept:A number is rational if it can be written as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.Explanation:Assume $\log_2 7$ is rational.Let $\log_2 7 = \frac{p}{q}$, where $p,q \in \mathbb{N}$.By the definition of logarithm, $2^{p/q}=7$.Raising both sides to the power $q$ gives $2^p=7^q$.The left side $2^p$ is even, while the right side $7^q$ is odd.An even number cannot equal an odd number.So $\log_2 7$ is not rational.It is also not an integer, since $2^n=7$ has no integer solution.Hence it is not a prime number.Therefore, $\log_2 7$ is irrational.Answer:$\log_2 7$ is an irrational number.