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Resolució d'equacions exponencials 1

Veiem com es resolen els tipus d'equacions exponencials que treballarem en aquest curs.


Exemples 

bold 2 to the power of bold x bold equals bold 32

             2 to the power of x equals 2 to the power of 5 space space space space rightwards double arrow space bold italic x bold equals bold 5

bold 9 to the power of bold x bold equals bold 1 over bold 27 open parentheses 3 squared close parentheses to the power of bold x equals 1 over 3 cubed 3 to the power of 2 x end exponent equals 3 to the power of negative 3 end exponent space space space space rightwards double arrow space space 2 x equals negative 3 space space rightwards double arrow space space bold space bold italic x bold equals bold minus bold 3 over bold 2  

bold 8 to the power of bold minus bold x bold plus bold 10 end exponent bold equals bold 64 D e s c o m p o s e m space space e l space 64 space i space e l space 8 space 64 table row cell table row 64 2 row 32 2 row 16 2 row 8 2 row 4 2 row 2 2 row 1 blank end table end cell end table 64 equals 2 to the power of 6 i space j a space s a b e m space q u e space 2 cubed comma space l l a v o r s colon space q u e d a space d o n c s space colon space space space space space space space space space space space space space left parenthesis 2 cubed right parenthesis to the power of negative x plus 10 end exponent equals 2 to the power of 6 space space space space space space space space space space space space space space 2 to the power of 3 times left parenthesis negative x plus 10 right parenthesis end exponent equals 2 to the power of 6 I g u a l e m space e l s space e x p o n e n t s space j a space q u e space l e s space d u e s space p o t è n c i e s space t e n e n space l a space m a t e i x a space b a s e space space space space space space space space space space space space space space 3 times left parenthesis negative x plus 10 right parenthesis equals 6 space space space space space space space space space space space space space space space space space space minus 3 x plus 30 equals 6 space space space space space space space space space space space space space space space space space space space space minus 3 x equals negative 24 space space space space rightwards arrow space space x equals fraction numerator negative 24 over denominator negative 3 end fraction equals bold 8 space space space space space space space space space space space space space

bold 3 to the power of bold x bold equals fraction numerator bold 1 over denominator square root of bold 27 end fraction

3 to the power of x equals 1 over open parentheses 3 cubed close parentheses to the power of 1 divided by 2 end exponent 3 to the power of x equals 1 over 3 to the power of begin display style bevelled 3 over 2 end style end exponent equals 3 to the power of negative 3 divided by 2 space end exponent space space space space rightwards double arrow space space bold italic x bold equals bold minus bold 3 over bold 2


bold 2 bold times bold 25 to the power of bold x bold equals bold 10 bold times bold 5 to the power of bold x bold minus bold 2 end exponent up diagonal strike 2 times open parentheses 5 squared close parentheses to the power of x equals up diagonal strike 2 times 5 times 5 to the power of x minus 2 end exponent space space space space space space space space 5 to the power of 2 x end exponent equals 5 to the power of 1 plus x minus 2 end exponent space space space space space space space space space 5 to the power of 2 x end exponent equals 5 to the power of x minus 1 space space end exponent rightwards double arrow space space 2 x equals x minus 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 2 x minus x equals negative 1 space rightwards arrow space space space space bold italic x bold equals bold minus bold 1

Observació important:

fixeu-vos que en aquest exemple i en tots els exemples anteriors sempre podem expressar els termes com a potencies d'un mateix nombre. O sigui, sempre arribem a una igualtat de potencies de la mateixa base.  En l'exemple anterior:

       space space space space space space space space space 5 to the power of 2 x end exponent equals 5 to the power of x minus 1 space space end exponent

i el raonament que fem és "igualtat de potencies de la mateixa base, implica igualtat d'exponents": 

                 space space space space space space space space space 5 to the power of 2 x end exponent equals 5 to the power of x minus 1 space space end exponent rightwards double arrow 2 x equals x minus 1

Què fem si no podem expressar els termes en la mateixa base?

Per exemple:   2 to the power of x equals 5 

En aquest cas hem d'aplicar logaritmes. Però això ho veurem en els apartats posteriors quan estudiem logaritmes.  


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