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?
NoteIn 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.