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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Use indirect reasoning to prove that log2 5 is an irrational number.

To determine

To prove: log25 is an irrational number

Explanation

Definition used:

Logarithmic function

A function such as y=logax or y=lnx that is inverse of an exponential function such as y=ax or y=ex so that the independent variable appears in a logarithm

Proof:

Suppose log25 is rational number (1)

Rational number is the form pq,q0 where p,q is a natural number.

log25=pq, for natural numbers p and q.

By definition of logarithmic function,

2pq=52p=5q

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

No two natural numbers satisfy above equation.

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

Thus, log25 is an irrational number.

Hence the result.

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