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Simplificación de expresiones/ Descomposición en fracciones simples/ log((sqrt(x^2+2)-x*sqrt(3))/(sqrt(x^2+2)+x*sqrt(3)))/log(10)

¿Cómo vas a descomponer esta log((sqrt(x^2+2)-x*sqrt(3))/(sqrt(x^2+2)+x*sqrt(3)))/log(10) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   /   ________          \
   |  /  2            ___|
   |\/  x  + 2  - x*\/ 3 |
log|---------------------|
   |   ________          |
   |  /  2            ___|
   \\/  x  + 2  + x*\/ 3 /
--------------------------
         log(10)
log⁡(−3x+x2+23x+x2+2)log⁡(10)\frac{\log{\left(\frac{- \sqrt{3} x + \sqrt{x^{2} + 2}}{\sqrt{3} x + \sqrt{x^{2} + 2}} \right)}}{\log{\left(10 \right)}} 
log((sqrt(x^2 + 2) - x*sqrt(3))/(sqrt(x^2 + 2) + x*sqrt(3)))/log(10)
Respuesta numérica [src] 
0.434294481903252*log((sqrt(x^2 + 2) - x*sqrt(3))/(sqrt(x^2 + 2) + x*sqrt(3)))
0.434294481903252*log((sqrt(x^2 + 2) - x*sqrt(3))/(sqrt(x^2 + 2) + x*sqrt(3)))
Denominador común [src] 
   /        ________                             \
   |       /      2                    ___       |
   |     \/  2 + x                 x*\/ 3        |
log|--------------------- - ---------------------|
   |   ________                ________          |
   |  /      2        ___     /      2        ___|
   \\/  2 + x   + x*\/ 3    \/  2 + x   + x*\/ 3 /
--------------------------------------------------
                     log(10)
log⁡(−3x3x+x2+2+x2+23x+x2+2)log⁡(10)\frac{\log{\left(- \frac{\sqrt{3} x}{\sqrt{3} x + \sqrt{x^{2} + 2}} + \frac{\sqrt{x^{2} + 2}}{\sqrt{3} x + \sqrt{x^{2} + 2}} \right)}}{\log{\left(10 \right)}} 
log(sqrt(2 + x^2)/(sqrt(2 + x^2) + x*sqrt(3)) - x*sqrt(3)/(sqrt(2 + x^2) + x*sqrt(3)))/log(10)
Denominador racional [src] 
   /                 ________    ________              ________              ________\
   |        2       /      2    /  2            ___   /      2        ___   /  2     |
   |     3*x      \/  2 + x  *\/  x  + 2    x*\/ 3 *\/  2 + x     x*\/ 3 *\/  x  + 2 |
log|- --------- - ----------------------- + ------------------- + -------------------|
   |          2                  2                       2                     2     |
   \  -2 + 2*x           -2 + 2*x                -2 + 2*x              -2 + 2*x      /
--------------------------------------------------------------------------------------
                                       log(10)
log⁡(−3x22x2−2+3xx2+22x2−2+3xx2+22x2−2−x2+2x2+22x2−2)log⁡(10)\frac{\log{\left(- \frac{3 x^{2}}{2 x^{2} - 2} + \frac{\sqrt{3} x \sqrt{x^{2} + 2}}{2 x^{2} - 2} + \frac{\sqrt{3} x \sqrt{x^{2} + 2}}{2 x^{2} - 2} - \frac{\sqrt{x^{2} + 2} \sqrt{x^{2} + 2}}{2 x^{2} - 2} \right)}}{\log{\left(10 \right)}} 
log(-3*x^2/(-2 + 2*x^2) - sqrt(2 + x^2)*sqrt(x^2 + 2)/(-2 + 2*x^2) + x*sqrt(3)*sqrt(2 + x^2)/(-2 + 2*x^2) + x*sqrt(3)*sqrt(x^2 + 2)/(-2 + 2*x^2))/log(10)
Combinatoria [src] 
   /        ________                             \
   |       /      2                    ___       |
   |     \/  2 + x                 x*\/ 3        |
log|--------------------- - ---------------------|
   |   ________                ________          |
   |  /      2        ___     /      2        ___|
   \\/  2 + x   + x*\/ 3    \/  2 + x   + x*\/ 3 /
--------------------------------------------------
                     log(10)
log⁡(−3x3x+x2+2+x2+23x+x2+2)log⁡(10)\frac{\log{\left(- \frac{\sqrt{3} x}{\sqrt{3} x + \sqrt{x^{2} + 2}} + \frac{\sqrt{x^{2} + 2}}{\sqrt{3} x + \sqrt{x^{2} + 2}} \right)}}{\log{\left(10 \right)}} 
log(sqrt(2 + x^2)/(sqrt(2 + x^2) + x*sqrt(3)) - x*sqrt(3)/(sqrt(2 + x^2) + x*sqrt(3)))/log(10)
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