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KCC's Quizzes AQQ253 about a funny equation

And here is a new challenge! It's easier than the previous one but still...

Please spread those weekly quizzes among your colleagues!

Also, please be informed awards and goodies are being sent to the frequent winners. Please apologize for the long delay since we do that only 2 times per year!

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Some statistics about the frequent winners by country origins: (we have also statistics by individuals, but they can be provided on demand and only with own data).

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Kuo-Chang

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Parents
  • 8^x + 2^x = 130

    Assume 2^x = t then the equation will become t^3+t-130 = 0

    So possible value of t will be 5 so then equation will be (t - 5) (t^2+5t+26) = 0.

    And t^2+5t+26 will give us imaginary roots because discrimant is 5*5-4*1*26 which is negative so possible value of t is 5 

    Hence value of x is log5/log2 which is 2.3219

    Ans 2.3219

  • Thnaks  fr your prompt feedback!

    Yes, 2.3219 is indeed a solution to the equation. BUT there are others! The question was saying all the possible solutions for x (not necessarily real numbers!). may be you can retreive them?

Reply Children
  • So we can get value of t as (-5+i√79) /2 and  (-5-i√79) /2 so value of x will be log( -5+i√79) /2*log2 and  (-5-i√79) /2*log2.

    Not sure if we can calculate further. 

    If we can please help me know more about that. 

  • Use the Euler transformation:   a + b i   =  r*exp(i*t)    with r^2  = a^ + b^2   and t is the angle with a base "a" and a rise "b". 
                                                     
    So  ln(a+bi)  =  ln( r*exp(it)) = ln(r) + ln(exp( i*t ))  =  ln(r)  +  i*t  

    (Euler transform is like to transform a point to polar coordinate  from its cartesian coordinates (real and imaginary axis))

  • Yes! @Sina22! You get it. The expressions fo the 2 complex values can be simplified as -2.5 +/- i 4.4444

    Congratulation!