Is ln sqrt(2) irrational? I know that the natural log of any positive algebraic number is transcendental, as a consequence of the Lindemann-Weierstrass theorem, but what about the natural log of the square root of two (which is irrational). Is this rational or irrational?

zapri4j 2022-09-17 Answered
Is ln 2 irrational?
I know that the natural log of any positive algebraic number is transcendental, as a consequence of the Lindemann-Weierstrass theorem, but what about the natural log of the square root of two (which is irrational).
Is this rational or irrational?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

berzamauw
Answered 2022-09-18 Author has 9 answers
Not only is ln ( 2 ) irrational, but it's also transcendental!
Proof:
ln ( 2 ) = ln ( 2 1 / 2 ) = 1 2 ln ( 2 ) T
which is transcendental.
For reference, T is the set of transcendental numbers.

We have step-by-step solutions for your answer!

Makaila Simon
Answered 2022-09-19 Author has 2 answers
If you already know that the log of a positive algebraic number is transcendental, then all you need to realize is that 2 is a positive algebraic number. 2 is a root of x 2 2 = 0
Therefore, log ( 2 ) is transcendental log ( 2 ) is irrational.

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-06-07
ransform a formula for doubling time
I have a problem with transforming a formula for doubling time. It has two variables: increase (i) and number of periods (p).
The original formula: p = log ( 2 ) log ( 1 + i )
When I enter i = 0.05 it correctly calculates p = 14.207
Now I'm trying to make a formula for "increase". I transformed it to: i = 10 log ( 2 ) p 1.. When I enter 14.207 periods, it gives me a increase = 0.1189 (and not 0.05). Where's my mistake?
The transformations I did:
p = log ( 2 ) log ( 1 + i )
log ( 1 + i ) = log ( 2 ) p
1 + i = 10 log ( 2 ) p
i = 10 log ( 2 ) p 1
It gives a wrong answer... Do you see where I made a mistake?
asked 2022-01-20
How do I find the common logarithm of a number?
asked 2022-07-05
Solving this logarithm equation?
How do I solve this equation using common logarithms?
log x = 1 log ( x 3 )
asked 2021-10-24
Solve the given equation for x 2=e0.06x
asked 2022-04-25
How can we solve:
x+ln(x)1=0?
asked 2022-09-04
Find the smallest positive integer k such that the baseten representation of 3k has exactly one hundred digits.
If log 8 3 = p   a n d   log 3 5 = q , then, in terms of p and q, log 5 equals to?
asked 2022-06-15
Confusion between negative and positive signs in natural logarithm
If z = 0.1887 ( x 0.7637 ) ( y 0.2306 )
Its natural logarithm will be
ln ( z ) = [ ln ( 0.1887 ) + 0.7637 ln ( x ) + 0.2306 ln ( y ) ]
or
ln ( z ) = [ ln ( 0.1887 ) 0.7637 ln ( x ) 0.2306 ln ( y ) ]?
Thank you in advance