# Solving a system of 4 logarithmic equations? [closed]

I would like to solve this system of 4 logarithmic equations in Mathematica.

• a*log(b+d)=20
• a*log(b*30^c+d)=125
• a*log(b*180^c+d)=710
• a*log(b*360^c+d)=1350

I am looking for the values of a, b, c, and d.

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## closed as off-topic by Szabolcs, ciao, Yves Klett, gpap, george2079Mar 26 '14 at 14:58

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Is this a question about a mathematics problem or about how to use the software product Mathematica? –  Michael E2 Mar 26 '14 at 0:07

There are no solutions as shown by Solve:

{eq1, eq2, eq3, eq4} = {
a*Log[b + d] == 20,
a*Log[b*30^c + d] == 125,
a*Log[b*180^c + d] == 710,
a*Log[b*360^c + d] == 1350
};

Solve[{eq1, eq2, eq3, eq4}, {a, b, c, d}]
(* {} *)


To show why there are no solutions over the reals, let's investigate a little further. First let's define our own Solve that won't introduce any arbitrary parameters when taking logarithms.

solve[e__] := Normal[Solve[e] /. _C -> 0, ConditionalExpression]


Now let's eliminate a and c via substitution then use ContourPlot to see where root candidates are for b and d. (After playing around with the plot range, I could only find one apparent intersection.)

Module[{A, C},
A = First@solve[eq1, a];
C = First@solve[eq2 /. A, c];
ContourPlot[Evaluate[{eq3, eq4} /. A /. C], {b, -.01, .01}, {d, -.01 + 1, .01 + 1}, MaxRecursion -> 5]
]


So it looks like b == 0 and d == 1 are candidates for roots. But subbing into our original system shows these are extraneous roots.

{eq1, eq2, eq3, eq4} /. {b -> 0, d -> 1}
(* {False, False, False, False} *)

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{eq1, eq2, eq3, eq4} = {a*Log[b + d] == 20, a*Log[b*30^c + d] == 125,
a*Log[b*180^c + d] == 710, a*Log[b*360^c + d] == 1350};


and

FindRoot[{eq1, eq2, eq3,
eq4}, {{a, 0.1}, {b, 0.1}, {c, 0.01}, {d, 0.01}}]

(*  {a -> 1.43923, b -> 3115.74, c -> -1.15142, d -> 19.1192}  *)


?

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I get a warning, FindRoot::cvmit: Failed to converge to the requested accuracy or precision within 100 iterations. And substituting the values into the equations gives $11.5863 = 20$, $6.32768 = 125$, $4.74367 = 710$, $4.49181 = 1350$. –  Rahul Mar 26 '14 at 9:09
@Rahul Narain I did not. You can try to look for a better starting point and also play with the option Method –  Alexei Boulbitch Mar 26 '14 at 13:56
I'm just curious why you posted an answer that finds a non-solution eight hours after RiemannZeta already proved that there are no solutions. –  Rahul Mar 26 '14 at 19:10
@ Rahul Narain Apologies, I did not check whether or not the results indeed deliver a solution. As to your question, FindRoot is the right tool to solve such a system of equations, which is actually my message. The fact that it does not converge to a solution is a strange fact in itself. Usually such things are to be reported as bug suspects. As to your question about timing, my aim was to help you, and it is not important 8 hours earlier or later. I do not believe that here we look for priorities. –  Alexei Boulbitch Mar 27 '14 at 8:46