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[burakkucat]

Some weeks ago, I wrote a simple equation at the top of a sheet of A4 paper, where y was expressed as a function of x --

y = 10 log x                                            [Eqn. 1]

For no reason whatsoever, I decided to rearrange it such that x was expressed as a function of y --

x = 10^(y/10)                                              [Eqn. 2]

Plotting the curve of [Eqn. 1] (or [Eqn. 2]) showed the expected result.

Interchanging the x & y symbols in [Eqn. 2] gave a new, but similar looking, equation --

y = 10^(x/10)                                              [Eqn. 3]

Clearly a plot of that curve should be equivalent to the plot of [Eqn.2] (or [Eqn. 1]) rotated by 180 degrees around an axis defined by the function --

y = x                                                     [Eqn. 4]

Plotting the curve of [Eqn. 3] showed the expected result.

As the symmetry was pleasing, both [Eqn. 1] and [Eqn. 3] were plotted on the same chart.

Those two curves intersect at two points; the coordinates of first point of intersection are not an integer whereas the coordinates of the second point of intersection are 10 & 10.

Adding the plot of [Eqn. 4] to the chart shows all three curves intersecting at the same two points.

Looking a little closer, we see that the coordinates of the first point of intersection are between 1.25 & 1.25 and 1.50 & 1.50.

It should be possible to calculate the two points of intersection. Substituting [Eqn. 4] into [Eqn. 1] gives --

x = 10 log x

and rearranging gives --

x / (log x) = 10                                        [Eqn. 5]

Likewise, substituting [Eqn. 4] into [Eqn. 3] gives --

x = 10^(x/10)

and rearranging gives --

x / (10^(x/10)) = 1                                       [Eqn. 6]

Finally, substituting [Eqn. 3] into [Eqn. 1] gives --

10^(x/10) = 10 log x

and rearranging gives --

(10^(x/10)) / (log x) = 10                              [Eqn. 7]

Solving either [Eqn. 5], [Eqn. 6] or [Eqn. 7] should give two values for x and substituting those two values (of x) into [Eqn. 4] would give the two corresponding values of y.

It is at this point that I got stuck. From inspection of either [Eqn. 5], [Eqn. 6] or [Eqn. 7] I could see one value --

x = 10

but not the second value.

Looking at the fifth chart (above) for the lower & upper limits and plugging those values into [Eqn. 5] and solving for each, by usage of a Casio fx-570MS calculator, via a classic "binary chop" showed that the first intersection was at coordinates 1.37128857 & 1.37128857 (to eight decimal places).

So, finally, my query for the mathematicians: How do I solve either [Eqn. 5], [Eqn. 6] or [Eqn. 7] to obtain the two values of x? 


Note

In the above, I have used log x to represent log₁₀ x.

Confession.

Many years ago, at the age of 16 I gained a GCE O level in Mathematics, at the age of 17 I gained a GCE O level in Additional Mathematics, at the age of 18 I gained a GCE A level in Pure Mathematics and at the age of 19 I completed a first year undergraguate subsidary course in Mathematics. 

[renluop]

....and then you self identified as feline?

[22over7]

Interesting. If you fool around with your equations a bit more, you can get

    x = log ((log x¹⁰)¹⁰)

But I can't see a "closed form" solution (other than 10).

Googling about for an idea, I came across https://arxiv.org/abs/math/9805045, that might shed some light
on this kind of "transcendental" equation.  Weirdly, an open problem ("Schanuel's conjecture") is involved.

But maybe there is some neat solution...

[Ronski]

Many years ago I gained a Grade 4 CSE in Mathematics..................so I don't have a clue as to what your on about 

[burakkucat]

. . . I can't see a "closed form" solution (other than 10).

Thank you for considering those equations. So I haven't forgotten everything . . .

Googling about for an idea, I came across https://arxiv.org/abs/math/9805045, that might shed some light
on this kind of "transcendental" equation.  Weirdly, an open problem ("Schanuel's conjecture") is involved.

That is interesting.

In a "mad moment", I was thinking about looking at the first derivative . . . but then decided against it.

[22over7]

If one googles "1.37128857", pretty soon one is led to https://arxiv.org/abs/1905.10438,
a (translation from Latin of a textbook) by Euler from 1755, where the problem he set himself (bottom page 18) is:

To find a number, other than10,  whose tabulated  logarithm  becomes equal to the tenth part of the number itself.

It's remarkable that our Esteemed burakkucat has stumbled (unknowingly??) into the (vicinity of) Euler's footsteps....
It's not easy to make out what is going on in Euler's text, but he seems to do what burakkucat does, and bash out
his solution by an approximation.
 

[burakkucat]

It's remarkable that our Esteemed burakkucat has stumbled (unknowingly??) into the (vicinity of) Euler's footsteps....

Thank you for the accolade. 

I had no idea of the above, as I am just a retired scientist who is rapidly forgetting all mathematics that was learnt.

With a forum username of 22over7 I am not surprised with your area of knowledge. (For an approximation of that particular constant, I would use "355over113" . . .    )

[burakkucat]

As my query, posed in the OP, is so close to Leonhard Euler's example in his paper "On the Use of Differential Calculus in the Resolution of Equations"*, I have attached a copy of the two relevant pages (as a PDF file), below.


*Original title: "De Usu Calculi Differentialis in Aequationibus resolvendis", first published as part of the book Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum, 1755, reprinted in Opera Omnia: Series 1, Volume 10, pp. 422 - 445, Eneström-Number E212, translated by: Alexander Aycock for the "Euler Kreis Mainz".