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Question: The logarithmic equation

Question:The logarithmic equation

MichaelVio 50 Maple 2021
optimization logarithmic
Yesterday at 12:44 PM
1

Please advise. I have a logarithmic equation.

   E(ν)=E0·ln(1+2·A·ν)+Em+C·ν^2

With constant E0>0, Em<0, C<0 and A>0, and E(ν) of variable ν between [10^4 and 10^13]

I have 3 points, ν1, ν2, and ν3, and the value of the function E(ν) at each point. I want to choose A to obtain the smallest absolute value of Em, preferably ~ -0.00005. The value of ν1, ν2, ν3 and E(ν1), E(ν2), E(ν3) are known.

With what “A” should I start to obtain  Em~ -0.00005?

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restart;
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Digits:=25;

25

 (1)
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E(nu):=Em+E0*ln(1+A*3.7653333333333333333*10^(-18)*nu)+ C*nu^2;

Em+E0*ln(1+0.3765333333333333333300000e-17*A*nu)+C*nu^2

 (2)
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E1:=subs(nu=5.754173859*10^8,E(nu));

Em+E0*ln(1+0.2166638263708799999980819e-8*A)+331105167995989518.8100000*C

 (3)
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E2:=subs(nu=6.338671958*10^8,E(nu));

Em+E0*ln(1+0.2386721281252266666645538e-8*A)+401787621911355537.6400000*C

 (4)
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E3:=subs(nu=31*10^9,E(nu));

Em+E0*ln(1+0.1167253333333333333323000e-6*A)+961000000000000000000*C

 (5)
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solve({E1=3.259,E2=3.59});

{A = A, C = 0.1414778271842943053715718e-16*E0*ln(1.+0.2166638263708799999980819e-8*A)-0.1414778271842943053715718e-16*E0*ln(1.+0.2386721281252266666645538e-8*A)+0.4682916079800141507799027e-17, E0 = E0, Em = 4.684403973756333877677291*E0*ln(1.+0.2386721281252266666645538e-8*A)-5.684403973756333877677291*E0*ln(1.+0.2166638263708799999980819e-8*A)+1.708462284686653486488817}

 (6)
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sol:=solve({E1=3.259,E2=3.59,E3=0},{Em,E0,C,A});

{A = A, C = .1000000000000000000000000*(3259.*ln(1.+0.2386721281252266666645538e-8*A)+331.*ln(1.+0.1167253333333333333323000e-6*A)-3590.*ln(1.+0.2166638263708799999980819e-8*A))/(7068245391536601883.*ln(1.+0.1167253333333333333323000e-6*A)-96066889483200401048119.*ln(1.+0.2386721281252266666645538e-8*A)+96059821237808864446236.*ln(1.+0.2166638263708799999980819e-8*A)), E0 = -31821175830670350532464.09/(7068245391536601883.*ln(1.+0.1167253333333333333323000e-6*A)-96066889483200401048119.*ln(1.+0.2386721281252266666645538e-8*A)+96059821237808864446236.*ln(1.+0.2166638263708799999980819e-8*A)), Em = 0.2000000000000000000000000e-2*(6037915335175266232043.*ln(1.+0.1167253333333333333323000e-6*A)-0.1565949500000000000000000e27*ln(1.+0.2386721281252266666645538e-8*A)+0.1724995000000000000000000e27*ln(1.+0.2166638263708799999980819e-8*A))/(7068245391536601883.*ln(1.+0.1167253333333333333323000e-6*A)-96066889483200401048119.*ln(1.+0.2386721281252266666645538e-8*A)+96059821237808864446236.*ln(1.+0.2166638263708799999980819e-8*A))}

 (7)
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A:=4.19747*10^6;evalf(sol[4]);#Em

4197470.00000

  

Em = -0.7291335370248713771647314e-1

 (8)
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Em:=-0.07291335370248713771647314;

-0.7291335370248713771647314e-1

 (9)
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evalf(sol[3]);#E0

E0 = 373.7014139352964920547905

 (10)
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E0:=373.7014139352964920547905;

373.7014139352964920547905

 (11)
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evalf(sol[2]);#C

C = -0.1549823289304468027271168e-18

 (12)
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C:=-1.549823289304468027271168*10^(-19);

-0.1549823289304468027271168e-18

 (13)
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E(nu):=Em+E0*ln(1+A*3.7653333333333333333*10^(-18)*nu)+ C*nu^2;

-0.7291335370248713771647314e-1+373.7014139352964920547905*ln(1+0.1580487370666666666652675e-10*nu)-0.1549823289304468027271168e-18*nu^2

 (14)
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plot(E(nu),nu=10^6..3.1*10^10);

 
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evalf(subs(nu=5.754173859*10^8,E(nu)));evalf(subs(nu=6.338671958*10^8,E(nu)));evalf(subs(nu=6.974105195*10^8,E(nu)));

3.258999999999999999999993

  

3.589999999999999999999993

  

3.948288407313424284054632

 (15)
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evalf(subs(nu=0,E(nu)));evalf(subs(nu=11.4*10^6,E(nu)));evalf(subs(nu=15.9*10^9,E(nu)));#44.5eV @ 15.9Ghz

-0.7291335370248713771647314e-1

  

-0.560769863114429190697634e-2

  

44.52276318342970513246218

 (16)
 

 

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