Question

I need to prove that log5(3) is not a rational number.
I'm not quite sure of how to prove this...

Answer 1

I think that for log5(3) to be rational, lg(3)/lg(5) must also be a rational number

Answer 2

well,let
log5(3) = a
then 5^a = 3

simplifies the thinking now hmm ?

Answer 3

a=1

Answer 4

wel, if that is proof enough then it was quite easy

Answer 5

i dont think still its been proven or it'll be worthwhile carrying forward this way..
lets hear others..
@hba what do you mean by a=1 ?

Answer 6

I wanted to play a let game but it wont work here.

But we can see,

A rational number is a number which can be written as a fraction of two integers, where the denominator is nonzero. (Another way of saying the same thing: A rational number is a number that can be written in the form p/q, where p/q are integers and q ≠ 0.)


So it would be

a=log(3)+2i pi n/log(5)

Where n is a set of integers.

Answer 7

I do not know how can i prove it is not a rational number though :/

Answer 8

continuing what @shubhamsrg have done
well,let
log5(3) = a ..... (where a is a rational number)
then 5^a = 3

since, a is rational number, it can be expressed in the form a=b/c
where, b and c are INTEGERS.
so, we have.
\(\huge \sqrt[c]{5^b}=3 \implies5^b=3^c\)
now there are no pair of integers b,c which can satisfy this !
hence a cannot be rational number.
this is proof by contradiction.