Solve aB=a^B in Neutral Operations Using Algebra

This article shows how to solve a*b=a^b, which is a Neutral Operation since the operators are being held neutral to one another using the same variable values for a and b. Simple algebra will be used. If a*b represents Area, and a^b represents an elongated graceful exponential Curve, perhaps Mother Nature has a choice / tradeoff / "tipping point" when certain numbers are in this relation. Possibly so. Learn how to determine this relation in a few algebraic steps for your own usage and theoretical understanding.

Steps

  • Become familiar with the basic image of the concept, i.e. when a*B = a^B, then B^(1/(1-B)=a, or for x*Y=x^Y then x = y^(1 (y 1)))

The tutorial

  1. Create a new Excel worksheet and enter into cell A1 a*b = a^b, where "^" is Excel's symbol for exponentiation.
    • Enter into cell A2 ab/a = (a^b)/a
    • Enter into cell A3 b = a^(b-1)
    • Enter into cell A4 b^(1/(b-1)) = a; a has now been isolated and defined in terms of b and 1 but b may not = 1 lest division by 0 result in the denominator.
    • The simple algebra is complete.
  2. Substitute in trial values for b in order to obtain the value of a;
    • Let b = 10. Enter in cell C1 ="a=b^(1/(b-1))"
    • Enter into cell F1 the word "Var b" for variable b.
    • Enter into cell F2 the value 10.
    • Do MenuCommand Insert Name DefineName "Var" to cell F2.
    • Enter into cell C2 the formula, without quotes, "=Var^(1/(Var-1))"
    • Copy the value in cell C2 and do Edit Paste Special Value into cell C3.
    • The value you obtain should be 1.29154966501488
  3. Test the hypothesis of the original formula, i.e. that a*b = a^b:
    • Enter a*b into cell F3
    • Enter the formula, without quotes, "=C2*Var" into cell F4.
    • Enter a^b into cell F5
    • Enter the formula, without quotes, "=C2^Var" into cell F6.
    • Enter the formula, without quotes, "=F4-F6" into F7. The result should be 0. Done, because C2*Var = C2^Var, and the difference in F7 is 0!

Extra Credit Assignment

  1. It is left to the ingenuity of the reader to prove to themselves, that for Mother Nature, this equation represents a tradeoff or tipping point between area (multiplication) and curve extension (exponentiation) by:
    • Copying the above section to an area below it;
    • Doing Edit Replace Replace All a with x, b with y, c with z and Var with Var2;
    • Defining properly the Var2 cell with the Defined Name Var2 via Insert Name Define;
    • Creating a column of the given variable and the dependent formula and filling down the formula, then entering close values for the given variable, probably with an Edit Series menu command;
    • A little hard, but do-able: Create the x*y rectangular areas for a chart in terms of lines extending from {0, 0}. For example, if x*y (sub 1) in the first case = 200 and x*y (sub 2) = 202 in the 2nd case, you might create rectangle 1 that's 20*10 and rectangle 2 that's 20.5*(202/20.5=9.85365853658537), or rather, whatever works in the algebraic relation of a*b = a^b, certainly -- one variable should be incremented by an Edit Fill Series menu command is more to the point, and the other left to be computed by the neutral operation;
    • And then also charting x^y columnar data as a series on the same chart, by pasting in the data and editing the series to correctly reference the columns and rows desired on the worksheet.
  2. Then you will see Nature's "Tipping Point" or Tradeoff, given the equality of resources (numerical amounts of something x and something y).

Tips

  • It is generally a good idea to format INPUT CELLS always as the same color; typically, canary yellow was used in the past. Format Color for cell F2 Canary Yellow. Save the worksheet as "a*b = a^b, Neutral Operation".
  • It may be that a Neutral Operation numeric set represents a "tipping point" in Nature. In the above case, she may decide whether to create an area (by multiplying) or an exponential curve (such as an asymptotic stem of a plant, or a growth curve such as e^x).
  • n.b. A new Part, The Quadratic Relation of Neutral Operations / Symmetry by Commutation, has been added to the article How to Convert a Quadratic Formula to Roots Form by Completing the Square
  • Let E = mc^n as n --> 2. Then E/m = c^n = cn by the Neutral Operation described above. cn = c^n; cn/c = (c^n)/c; n= c^(n-1); n^(1/(n-1)) = c and if c = 186,100, then n = 5.37380552441777E-06 (done by goal seeking in XL). Or if c = 1, then n = 22,751,438,016,029,000 .. per goal seeking. It seems that the operator that produces n=2 is neither exponentiation or multiplication, in truth. "-->" means "approaches". Just for fun, 22,751,438,016,029,000*5.37380552441777E-06 = 122,261,803,299 and 22,751,438,016,029,000^5.37380552441777E-06 = 1.00020241629592 but

186100*5.37380552441777E-06=1.00006520809415=186100^5.37380552441777E-06 but 1*22,751,438,016,029,000 ≠ 1^22,751,438,016,029,000, which I cannot explain, as c*n should equal c^n. Probably for c=1, it simply will not work, so setting c (or c^2) equal to 1 is unjustified according to this. Light speed c has a definite, quantifiable velocity and c^2 a rate of acceleration, if I understand correctly. Perhaps I do not, however.

  • For n^(1/(n-1)) = c where n=2, we have 2^(1/1). Where n = 3/2, we have (3/2)^(1/(1/2))= 2.25, or E/m = c^(9/4) and so E/m > c^2, which is interesting. And for n = 5/4, we have (5/4)^(1/(5/4 - 1)) = 625/256 = 2.44140625 -- also interesting.

Helpful Guidance

  1. Make use of helper articles when proceeding through this tutorial:
    • See the Related Articles below and the article How to Do the Sub Steps of Neutral Operations for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation relating to Neutral Operations.
    • For more art charts and graphs, you might also want to click on Algebra, Mathematics, Spreadsheets or Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.

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