Chapter 1, Problem 14P

To determine

To prove: ${\log }_{2}5$ is an irrational number

Explanation of Solution

Definition used:

Logarithmic function

A function such as $y={\log }_{a}x$ or $y=\ln x$ that is inverse of an exponential function such as $y=a^x$ or $y=e^x$ so that the independent variable appears in a logarithm

Proof:

Suppose ${\log }_{2}5$ is rational number (1)

Rational number is the form $\frac{p}{q},q\ne 0$ where p,q is a natural number.

${\log }_{2}5=\frac{p}{q}$, for natural numbers p and q.

By definition of logarithmic function,

$\begin{array}{l}{2}^{\frac{p}{q}}=5\\ 2^p=5^q\end{array}$

The right hand side $2^p$ is even and the left hand side $5^q$ is odd.

No two natural numbers satisfy above equation.

Therefore, this is a contradiction to the assumption (1).

Thus, ${\log }_{2}5$ is an irrational number.

Hence the result.