Question

Evaluate the value of \[{\log _5}\left( {25} \right)\]

Answer 1

Hint: A logarithm can have any positive value as its base, but two log bases are more useful than the others. Logarithm is the inverse function 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”. Logarithm is a power to which a number must be raised in order to get some other number.

Complete step-by-step answer:
Rewriting as an equation according to the question we have,
 \[{\log _5}\left( {25} \right) = x\]
If “x” and “b” are positive real numbers and “b” does not equal \[1\] , then \[{\log _b}\left( x \right) = y\] is equivalent to \[{b^y} = x\] .
 Hence we can write,
 \[{5^x} = 25\]
Now expressions in the equation that all have equal bases are created
 \[{5^x} = {5^2}\]
Since, the bases are the same, and then two expressions are only equal if the exponents are also equal.
Therefore we have,
 \[x = 2\]
The variable \[x\] is equal to \[2\] .
So, the correct answer is “ \[x = 2\] ”.

Note: The logarithm is the inverse function 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”. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication. It is important to create equal bases for easy calculations since logarithmic scales reduce wide-ranging quantities to tiny scopes. Logarithm is a power to which a number must be raised in order to get some other number.