Euler's intuition (which later proved correct)
From
A.O. Gelfond
's classic
Transcendental and Algebraic Numbers of Numbers
(English translation of Russian original) I quote:
The EulerHilbert problem. The problem of the transcendence or the rationality of the logarithms, with rational base, of rational numbers, stated by Euler in 1748, was formulated by Hilbert in a significantly more general form and introduced by him as number seven of a set of 23 problems, to the solution of which there appeared to be no suitable approach even at the very end of the nineteenth century...
(I take up this topic in greater detail later.)
Euler introduced the concept of a transcendental number (When exactly? Was it done before Euler? Was it Leibnitz? Is there some confusion over the difference between transcendental
function
and transcendental
number
?), but he could
not
prove the transcendence of
even a single
example (one imagines that he must have wondered if (e.g)
or
e
were transcendental). Euler did however make a wonderful guess (Fel'dman and Shidlovskii  in their monumental 1967
Survey
wrote:
"...we may mention the conjecture made by Euler in 1748 (they give as reference Euler's Introductio in analysin infinitorum, Lausanne 1748, Opera omnia, VIII and IX) on the transcendence of the logarithms to a rational base of rational numbers that are not rational powers of the base."
) at some numbers that he felt could be transcendental (it is an elementary exercise that his candidates are all irrational), as I now explain.
Standard, routine high school problems are to ask:

Solve the equation
(later vary the pair (4, 8) to e.g: (4, 16), (
,
), (
, 10), ... )

Solve the equation
(later vary the pair (4, 6) to e.g: (5, 2), (10, 8), (
,
), ... )
Of course both equations have solutions, meaning:

there
is
an '
' such that

there
is
an '
' such that
But what
are
the values of
and
? School pupils  spotting the obviously related '4' and '8'  should get
to be
, but might get stuck with finding
. Some might say something like:
, then  using a calculator  write:
=
(to 10 places of accuracy)
It is entirely elementary to prove that
is, in fact, irrational. One might be astounded to know that
is
actually
transcendental.
This is what I believe Euler surmised
: let
and
be positive rational numbers such that
, then
p
is rational
OR
p
is transcendental
(Alternatively: the ratio of the logarithms of two rational numbers
is either rational or transcendental.)
One should sense the
remarkable nature
of this assertion/guess/conjecture... it is saying/suggesting that such a ratio is either
incredibly simple
or
incredibly complex
.
>
restart;
>
with(plots): # for 'display'
with(plottools): # for 'line'
p := plot(4^x, x = 0.9..1.6, colour=navy, thickness=2,
title="THE GRAPH OF 4 TO THE POWER x"):
v1 := line([1.5, 0], [1.5, 8], color=red, thickness=2):
alpha := log[10](6.0)/log[10](4.0):
v2 := line([alpha, 0], [alpha, 6], colour=brown, thickness=2):
tp1 := textplot([1.48, 8,`x[1]=3/2`],align=LEFT):
tp2 := textplot([1.51, 8,`y=8`],align=RIGHT):
tp3 := textplot([1.3, 6,`y=6`],align=RIGHT):
tp4 := textplot([1.28, 6.2,`Is x[2]=transcendental?`],align=LEFT):
display([p, v1, v2, tp1, tp2, tp3, tp4]);
Warning, the name changecoords has been redefined
>
Hilbert's seventh problem (later section) is a more general form of Euler's surmise.