Question

$\sf{The \: number \: of \: distinct \: quadruple \: (a,b,c,d) \: of \: rational}$
$\sf{numbers \: exists \: for \: a \: log_{10}(2) \: + \: b \: log_{10}(3) \: + \: c \: log_{10}(5) \: +}$
$\sf{d \: log_{10}(7) \: = \: 2017 \: is}$



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Answer 1

Answer:

$log_{10}2^a+log_{10}3^b+log_{10}5^c+log_{10}7^d=2017\\\\\implies log_{10}(2^a\times 3^b \times 5^c \times 7^d)=2017$

$2017=log_{10}10\times 2017\\\\\implies log_{10}10^{2017}$

Taking anti-log both sides in the original equation , we get :

$10^{2017}=2^a \times 5^c \times 7^d \times 3^b$

Now , $10^{2017}$ has no factors of 3 and 7 .

So we can say that b and d has to be 0 in the case . It is because anything to the power 0 is 1 .

Now we have to find the values of those a and c .

$10^{2017}=2^{2017}\times 5^{2017}$

From this we find a = 2017 , c = 2017

a and c = 2017 while b = d = 0 .

The number of quadruple is one and only one . This is the only quadruple :

( 2017 , 0 , 2017 , 0 ) .

$log_ab+log_ac=log_abc$

Comparing powers does not mean that we can do anything .

For instance : $a^n=b^n$ and then a = b if n is not 0 and 1 . So n can be 0 if the factors are not present in any of the sides .

$a^0=1$ where a is not 0 .

Answer 2

hope it helps................