Find the value of x in

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answer is A.

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Detailed Solution

We know that the logarithm is the inverse operation to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b must be raised, to produce that number x, which means

if then the logarithm denoted as log and calculated as

Here x, y, b are real numbers subjected to condition

and

We know that

We know the logarithmic identity involving power m≠0 where mm is real number as

We also know the logarithmic identity involving quotient as

We can change to base our choice k>0,k≠1 using the identity
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When the base of the logarithm b=e an transcendental irrational with approximate value e≃2.718then the logarithm is called natural logarithm and if b=10 then logarithm is called common logarithm.
The given expression is

We see that on the right hand side of the equation has two bases b=e,10. We use the base change identity (1) for x=16,b=10,k=e to have

We find the prime factorization of 16 in the above step and replace 16 as

in the above step. We similarly find prime factorization of 625 and replace 625 as

in the above step. We have

We use the logarithmic identity of power

for

in left hand side and for x=2,b=e,m=4 in the right hand side of the above step to have
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We divide both side of the equation by

because

and have,

We use the definition of logarithm and have,

Hence, option 1 is correct.