Animated Solution for Mathematics - Basic Mathematics: The number ${\log }_{2}7$ is

We are investigating the nature of the number ${\log }_{2}7$. While it appears simple, we must determine if it is an integer, a rational number, or an irrational number.

The Number Line

To locate ${\log }_{2}7$ on the number line, we examine the powers of 2. We know that $2^2=4$ and $2^3=8$.

Since 7 is trapped between 4 and 8, it follows that:

Because ${\log }_{2}4=2$ and ${\log }_{2}8=3$, we conclude that $2<{\log }_{2}7<3$. This confirms that the number is not an integer.

The Rational Assumption

To determine if the number is rational, we employ a Proof by Contradiction. We assume that ${\log }_{2}7$ is a rational number.

If it is rational, it must be expressible as a fraction $\frac{p}{q}$, where $p$ and $q$ are positive integers. We write:

The Exponential Leap

Using the definition of a logarithm, where ${\log }_{a}b=x$ implies $a^x=b$, we transform our equation into:

To eliminate the fraction in the exponent, we raise both sides to the power of $q$:

This simplifies to the following expression:

The Parity Clash

We now analyze the equation $2^p=7^q$. On the left side, $2^p$ is a power of 2, which is always an even number for any positive integer $p$.

On the right side, $7^q$ is a power of 7, which is always an odd number for any positive integer $q$. We have reached a logical impossibility where an even number must equal an odd number.

The Beauty of Truth

This contradiction proves that our initial assumption—that ${\log }_{2}7$ is rational—was fundamentally flawed.

Therefore, we conclude that ${\log }_{2}7$ is an irrational number. It is a value that cannot be expressed as a simple fraction, representing an infinite, non-repeating sequence of digits.