Question

If \[0 < x < 1\], then
$\begin{align}
  & a)0<{{\log }_{10}}x<1 \\
 & b){{\log }_{10}}x>1 \\
 & c){{\log }_{10}}x<0 \\
 & d){{\log }_{10}}x<-1 \\
 & e)\text{ none of these} \\
\end{align}$

Answer 1

Hint: Now first we will understand the concept of logarithm and relate the function with the exponent function. Now we will take the base of logarithm as 10 and check the behavior of the function for values ranging from 0 to 1. Hence we find the solution of the given equation.

Complete step by step solution:
Now first let us understand the concept of logarithm.
Logarithm is nothing but a function which gives the value of the power which must be raised to the base of logarithm to obtain a particular number.
Now let us first understand this with an example.
Consider the equation ${{10}^{2}}=100$. The same equation can be written in logarithmic form as ${{\log }_{10}}100=2$ where 10 is called base of logarithm.
Hence in general a log is written in the form ${{\log }_{b}}x$ where b is called the base of logarithm.
Also note here x is a positive real number.
Now consider taking the base of the function as 10.
Now we know that ${{\log }_{10}}1=0$ .
Now for all values of x greater than 1 the value of ${{\log }_{10}}x$ will be greater than 0.
Similarly for all values of x less than 1 the value of ${{\log }_{10}}x$ will be less than 0.
Hence we can say that for $0 < x < 1$ the value of ${{\log }_{10}}x < 0$

So, the correct answer is “Option C”.

Note: Now note that base of logarithm can be any positive value other than 1. Also if a is the base of logarithm then we have ${{\log }_{a}}a=1$. Suppose we have an exponent in the logarithm then we can simplify the expression with the help of identity ${{\log }_{b}}{{x}^{n}}=n{{\log }_{b}}x$ .