Lösungen

1

Remove["Global`*"]

N[563/7, 35]

Rationalize[
      80.428571428571428571428571428571428571428571428572]

a1 = 80.428571428571428571428571428571428571428571428572

10^6 a1

Round[10^6 a1 - a1]

Round[10^6 a1 - a1]/(10^6 - 1)

80428491/999999

8936499/111111

2

Remove["Global`*"]

Zur Lösung

Solve[{Abs[2 λ x]  + w == 0, 2  x + y == 14, 
          3  x + z == 56, z - w == 42}, {x, y, z, w}] // 
      Simplify

Sei μSignX der Ausdruck  μSignX = Abs[λ]*Sign[x].  Es gilt damit: μSignX*x = Abs[λ]*Sign[x]*x = Abs[λ*x].
Das Vorzeichen von  μSignX ist daher identisch mit dem Vorzeichen von x.  x existiert also nur, wenn  x>0 und μSignX > 0 ist, was für  0 < λ < 3/2 eintrifft. In allen anderen Fällen sind die Vorzeichen von x und μSign unterschiedlich.

Solve[{2 μSignX x  + w == 0, 2  x - y == 14, 
          3  x + z == 56, z - w == 42}, {x, y, z, w}] // 
      Simplify

x1[μSignX_] := (14/(3 - 2 μSignX))

Plot[x1[μSignX], {μSignX, -10, 10}, 
        PlotRange -> {-7, 7}];

Weitere Betrachtungen

Solve[{2 Abs[λ] Sgnx x  + w == 0, 
          2  x - y == 14, 3  x + z == 56, z - w == 42}, {x, 
          y, z, w}] // Simplify

Solve[{2 Abs[λ] Sign[x] x  + w == 0, 
          2  x - y == 14, 3  x + z == 56, z - w == 42}, {x, 
          y, z, w}] // Simplify

Solve[{2 Abs[λ] Sign[x] x  + w == 0, 
          2  x - y == 14, 3  x + z == 56, 
          z - w == 42}, {λ, y, z, w}] // Simplify

Solve[{2 μ Sign[x] x  + w == 0, 2  x - y == 14, 
          3  x + z == 56, z - w == 42}, {μ, y, z, w}] // 
      Simplify

μ[x_] := ((-14 + 3 x)/(2 x Sign[x]))

Plot[μ[u], {u, -10, 10}, 
        PlotRange -> {-3, 1}];

Nur der positive Bereich von μ brauchbar.

x[λ_] := (14/(3 - 2 Abs[λ]))

Plot[x[u], {u, -2, 2}];

Wenn μ= Abs[λ] Sign[x] und somit auch x positiv ist, sind die positiven x-Werte Lösung. Das kommt hier vor, wenn μ zwischen 0 und 3/2 = 1.5 liegt.

x[λ_] := (14/(3 - (-2 Abs[λ])))

Plot[x[u], {u, -10, 10}];

Wenn μ= Abs[λ] Sign[x] und somit auch x negativ ist, sind die negativen x-Werte Lösung. Das kommt hier nicht vor.

3

Remove["Global`*"]

4/Log[x]  Log[x, x/x^Log[2, x] a^Log[2, x]] - 
      8 Log[2, 4^Log[2, a]]  Log[2, 16]/Log[2, a] + 
      12 Log[2, Log[4, 16]^Log[2, 8]]

4/Log[x]  Log[x, x/x^Log[2, x] a^Log[2, x]] - 
        8 Log[2, 4^Log[2, a]]  Log[2, 16]/Log[2, a] + 
        12 Log[2, Log[4, 16]^Log[2, 8]] /. 
      Log[s__^t__] -> t Log[s]

(36 - ((32 Log[s^t])/Log)[2]) /. 
      Log[s_^t_] -> t Log[s]

36 - ((32 t Log[s])/Log)[2] /. {s -> 2, t -> 2}

((4 Log[(a^(Log[x]/Log[2])) (x^(1 - Log[x]/Log[2]))])/Log)[x]^2 /. Log[s__*t__] -> Log[s] +  Log[t]

((4 (Log[(a^(Log[x]/Log[2]))] + Log[(x^(1 - Log[x]/Log[2]))]))/Log)[x]^2 /. Log[s_^t_] -> t Log[s]

((4 (((Log[a] Log[x])/Log)[2] + Log[x] (1 - Log[x]/Log[2])))/Log)[x]^2 // Simplify

-28 + ((4 (Log[2] + Log[a] - Log[x]))/(Log[2] Log[x])) // 
      Simplify

4

Remove["Global`*"]

Solve[
      Sqrt[(1 - Log[10, Log[10, y]]) (1 + 
                Log[10, Log[10, y]])] == 0, {y}]

10000000000 // N

(10^(1/10)) // N

Solve[
        Sqrt[(1 - Log[10, Log[10, y]]) (1 + 
                  Log[10, Log[10, y]])] == 0, {y}] // N

Sqrt[(1 - Log[10, Log[10, y]]) (1 + Log[10, Log[10, y]])] /. 
      y -> (10^(1/10))

Sqrt[(1 - Log[10, Log[10, y]]) (1 + 
              Log[10, Log[10, y]])] /.   Log[x_] -> 
        log[x]

(Sqrt[(1 - log[Log[y]/Log[10]]/log[10]) (1 + log[Log[y]/Log[10]]/log[10])]) /. Log[x_] -> log[x]

(Sqrt[(1 - Log[10, Log[10, y]]) (1 + Log[10, Log[10, y]])] /. 
            y -> (10^10)) // N // Chop

Sollte 0 geben, wird aber bei Vorauseinstellung der Genauigkeit nicht exakt.

N[(Evaluate[
              Sqrt[(1 - Log[10, Log[10, y]]) (1 + 
                      Log[10, Log[10, y]])]] /. y -> (10^10)), 
        30] // Chop

5

Remove["Global`*"]

Solve[Log[u]^2 + Log[Sqrt[u]] - 9 == Log[E u], {u}]

(Log[u]^2 + Log[u^(1/2)] - 9 == 
          Log[E u]) /. {Log[s_^t_] -> t Log[s], 
        Log[s_ t_] -> Log[s] + Log[t]}

-9 + Log[u]/2 + Log[u]^2 == 1 + Log[u] /. 
      Log[u] -> v

solv = (Solve[-9 + (v/2) + (v^2) == 1 + v, {v}] // 
          Flatten)

v1 = v /. solv[[1]]

v2 = v /. solv[[2]]

Solve[Log[u] == v1, {u}]

Solve[Log[u] == v1, {u}] // N

Solve[Log[u] == v2, {u}]

Solve[Log[u] == v2, {u}] // N

((Log[u]^2 + Log[Sqrt[u]] - 9 == Log[E u]) // N) /. 
      u -> (E^((1/4) - ((Sqrt[161])/4)))

((Log[u]^2 + Log[Sqrt[u]] - 9 == Log[E u]) // N) /. 
      u -> (E^((1/4) + ((Sqrt[161])/4)))

6

Remove["Global`*"]

Sin[x 17^(1 - 5^0)]^2 + E^(Sin[3 Pi]) == 
      11011/100000 - Cos[x]^2

Solve[
      Sin[x 17^(1 - 5^0)]^2 + E^(Sin[3 Pi]) == 
        11011/100000 - Cos[x]^2, {x}]

7

Remove["Global`*"]

4^(2 (x - 1)) 6^(3  x) 8^(-1) // 
      Simplify

3^(4 - 2  x) 4^(3 - 3  x) // Simplify

27^x  128^x  (4^3)^x 9^
          x ==  4^3  9^2  128

(4^(2 (x - 1)) 6^(3  x) 8^(-1) == 
            3^(4 - 2  x)  4^(3 - 3  x)) // Simplify // 
      ExpandAll

(27 *128 * 4^3* 9)^x  ==  128^1* 4^3*9^2

Solve[(1990656^x) == 663552, {x}]

{{x -> Log[663552]/Log[1990656]}} // Flatten // 
      Simplify

Solve[(1990656^x) == 663552, {x}] // N

(4  Log[3] + 13 Log[2])/(5 Log[3] + 13 Log[2]) // 
      N

8

Remove["Global`*"]

Solve[{y == (x - 2) (x + 3), y == x/2 + u}, {x, 
        y}]

Solve[{y == (x - 2) (x + 3), y == x/2 + u}, {x, 
          y}] /. u -> -97/16

(Solve[{y == (x - 2) (x + 3), y == x/2 + u}, {x, 
              y}] /. u -> -97/16) // N

-97/16 // N

9

Remove["Global`*"]

z[x_, 
        y_] := (y^(-2) - x^(-2))^(1/
              3) ((x - y)^9)^(1/3) x ;   z[x, y]

n[x_, 
        y_] := ( (((x^2 - y^2)^(1/3)  (y^(-1) - 
                      x^(-1)))^(1/2)) (x^3)^(1/
                2) (x - y)) ;   n[x, y]

z[x, y]/  n[x, y] // Simplify

(((x (((-(1/(x^2)) + (1/(y^2))))^(1/3)) ((((((x - y))^9)))^(1/3)))/((Sqrt[(x^3)]) (x - y) (Sqrt[(-(1/x) + (1/y)) ((((x^2) - (y^2)))^(1/3))]))) /. {x -> 3, y -> 2}) // N

z[x, y]  /. (a_^b_)^c_ -> a^(b c)

n[x, y]  /. (a_^b_)^c_ -> a^(b c)

u1 = 
      z[x, y]/n[x, y]  /. (a_^b_)^c_ -> a^(b c)

(1/x + 
          1/y) /. (1/a_ + 
            1/b_) -> (a + b)/(a b)

(((((((x^2 - y^2)/((x^2)  y^2))))^(1/3)) (((x - y))^2))/((Sqrt[x]) (Sqrt[(((x - y)/(x y))) ((((x^2) - (y^2)))^(1/3))])))

(((((x - y))^2) ((((((x^2) - (y^2))/((x^2) (y^2)))))^(1/3)))/((Sqrt[x]) (Sqrt[(((x - y) ((((x^2) - (y^2)))^(1/3)))/(x y))])))  /. (a_^b_)^c_ -> 
        a^(b c)

(((x - y))^2) ((((((x^2) - (y^2))/((x^2) (y^2)))))^(1/3)) /. (a_^2 - b_^2) -> (a - b) (a + b)

u1 = (((x - y))^(7/3)) (x + y)^(1/3) x^(-2/
              3)  y^(-2/3)

u2 = 
      x^(1/2) (x - y) ^(1/2) (x - y)^(1/6) (x + 
              y)^(1/6) x^(-1/2)  y^(-1/
              2)

u1/u2

((((((x - y))^(5/3)) (((x + y))^(1/6)))/((x^(2/3)) (y^(1/6)))) /. {x -> 3, y -> 2}) // N

(((((x - y))^(9/6)) (((x^2 - y^2))^(1/6)))/((x^(2/3)) (y^(1/6))))

((((((x - y))^(9/6)) (((x^2 - y^2))^(1/6)))/((x^(2/3)) (y^(1/6)))) /. {x -> 3, y -> 2}) // N