Question

The logarithm of 100 to the base 10 is
(a) 2
(b) 4
(c) 1
(d) 3

Answer 1

Hint: Use the fact that the value of the logarithm of ‘x’ to the base ‘y’ is ${{\log }_{y}}x$. Write 100 in terms of the exponential power of 10. Further, simplify the expression using the logarithmic formula ${{\log }_{x}}{{x}^{a}}=a$ to get the value of the given expression.

Complete step-by-step answer:
We have to calculate the value of the logarithm of 100 to the base 10.
We know that the value of the logarithm of ‘x’ to the base ‘y’ is ${{\log }_{y}}x$.
Substituting $x=100,y=10$ in the above expression, the value of the logarithm of 100 to the base 10 is ${{\log }_{10}}100$.
We will now write 100 in terms of the exponential power of 10. Thus, we have $100={{\left( 10 \right)}^{2}}$.
We can rewrite the expression ${{\log }_{10}}100$ as ${{\log }_{10}}100={{\log }_{10}}{{\left( 10 \right)}^{2}}$.
We know the logarithmic formula ${{\log }_{x}}{{x}^{a}}=a$.
Substituting $x=10,a=2$ in the above formula, we have ${{\log }_{10}}{{10}^{2}}=2$.
Thus, we have ${{\log }_{10}}100={{\log }_{10}}{{10}^{2}}=2$.
Hence, the value of the logarithm of 100 to the base 10 is 2, which is option (a).

Note: One must know that the logarithmic functions are the inverse of exponential functions. This means that the logarithm of a number ‘x’ is the exponent to which another fixed number; the base ‘b’ must be raised, to produce that number ‘x’. We observed that base 10 should be raised to power 2 to get 100, i.e., ${{\left( 10 \right)}^{2}}=100$. Logarithm counts the number of occurrences of the same factor in repeated multiplication.