Question

1.Prove that log 2 + log16/15 + 12 log 25/24 +7log81/80 = 1, (Assuming base 10 in each log)
2. Write in a simple term and simplify the 48th term of (x-1/2x)raise to power of 82

Answer 1

 (x - 1/2x)^82
  (r+1)th term  is :

${}^{82}C_{r}\ x^{r}\ (\frac{-1}{2x})^{82-r}$

So 48 th term  is  for  r = 47 :

${}^{82}C_{47}\ x^{47}\ (\frac{-1}{2x})^{82-47}\\\\=\ (-1)^{35}\ {}^{82}C_{47}\ x^{47-35}\ \frac{1}{2^{35}}\\\\=- \ \frac{82!}{47!\ 35!}\ x^{12}\ \frac{1}{2^{35}}\\\\=-5.177 * 10^{12}\ x^{12}\ \ approximately$

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perhaps there is some problem in the given question.  The result is not coming to be correct.  IS there some typing mistake ??


  Log 2 + Log 16 /15  + 12 Log 25 / 24  + 7 Log  81/80
=  Log 2  + Log (2⁴ / 3*5)  + 12  Log (5² / 3*2³)  +  7 Log (3⁴ / 2⁴*5)
=  Log 2  + 4 * Log 2 - Log 3 - Log 5    +  12 * 2 Log 5  - 12 * Log 3  - 12 * 3 Log 2
         + 7 *4 Log 3 - 7 * 4 Log 2 - 7 * Log 5
 = (1 + 4 - 36 - 28) Log 2 + (-1 -12+ 28 ) Log 3  + (-1 + 24 - 7 ) * Log 5
 = -59 Log 2 + 15 Log 3  + 16 Log 5
 =