To prove: is an irrational number
A function such as or that is inverse of an exponential function such as or so that the independent variable appears in a logarithm
Suppose is rational number (1)
Rational number is the form where p,q is a natural number.
, for natural numbers p and q.
By definition of logarithmic function,
The right hand side is even and the left hand side is odd.
No two natural numbers satisfy above equation.
Therefore, this is a contradiction to the assumption (1).
Thus, is an irrational number.
Hence the result.
Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!