Question

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The no. of roots of equation
$7 \times x^{ log_{5}2 } + {2}^{ log_{5}x } = 64$
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Answer 1

Step-by-step explanation:

Given :-

7×x^(log 2 (5)) + 2^(log x (5)) = 64

To find :-

The number of roots of the equation ?

Solution :-

Given equation is

7×x^(log 2 (5)) + 2^(log x (5)) = 64

It can be written as

7×2^(log x (5)) + 2^(log x (5)) = 64

Since a^(log x (b)) = x^(log a (b))

=> (7+1)×2^(log x (5)) = 64

=> 8×2^(log x (5)) = 64

=> 2^(log x (5)) = 64/8

=> 2^(log x (5)) = 8

=> 2^(log x (5)) = 2^3

If bases are equal then exponents must be equal

=> log x (5) = 3

We know that

log N (a) = x => a^x = N

=> log x (5) = 3

=> x = 5^3

=> x = 5×5×5

=> x = 125

The value of x = 125

Since the number of solutions for x in the given equation then

The number of roots = 1

Answer:-

The number of roots of the given equation is 1

Used formulae:-

• a^(log x (b)) = x^(log a (b))

• log N (a) = x => a^x = N

• log a^m = m log a

Where the letter in ( ) represents base