Question

If $\dfrac{1}{2} \leqslant {\log _{0.1}}x \leqslant 2$, then
A) maximum value of x is $\dfrac{1}{{\sqrt {10} }}$
B) x lies between $\dfrac{1}{{100}}$ and $\dfrac{1}{{\sqrt {10} }}$
C) minimum value of x is $\dfrac{1}{{10}}$
D) minimum value of x is $\dfrac{1}{{100}}$
E) maximum value of x is $\dfrac{1}{{100}}$

Answer 1

Logarithmic functions are the inverses of exponential functions. In Logarithms, the power is raised to some numbers (usually, base number) to get some other number. It is an inverse function of exponential function. Here in the given expression the base of the logarithm is 0.1, hence applying the properties and simplifying the terms of the logarithm function we can find the range of x.

Complete step-by-step solution:
The given expression is
$\dfrac{1}{2} \leqslant {\log _{0.1}}x \leqslant 2$
The expression can be written as:
$_{0.1}\dfrac{1}{2}{ \leqslant _{0.1}}{\log _{0.1}}x{ \leqslant _{0.1}}2$
Simplifying the functions, we get
$\sqrt {0.1} < x < {\left( {0.1} \right)^2}$
Hence, the range of x is
$\dfrac{1}{{\sqrt {10} }} \leqslant x \leqslant \dfrac{1}{{100}}$
Therefore, x lies between $\dfrac{1}{{100}}$ and $\dfrac{1}{{\sqrt {10} }}$

Hence, option B is the correct answer.

Additional information:
Rules of Logarithms
Logarithms are a very disciplined field of mathematics. They are always applied under certain rules and regulations.
Given that ${a^n} = b \Leftrightarrow {\log _a}b = n$, the logarithm of the number b is only defined for positive real numbers.
$\Rightarrow a > 0\left( {a \ne 1} \right),{a^n} > 0$
The logarithm of a positive real number can be negative, zero or positive. Logarithmic values of a given number are different for different bases. Logarithms to the base a 10 are referred to as common logarithms. When a logarithm is written without a subscript base, we assume the base to be 10. Logarithms to the base ‘e’ are called natural logarithms. The constant e is approximated as 2.7183. Natural logarithms are expressed as ln x which is the same as log e
The logarithmic value of a negative number is imaginary and the logarithm of any positive number to the same base is equal to 1.
${a^1} = a \Rightarrow {\log _a}a = 1$
The logarithm of 1 to any finite non-zero base is zero.
${a^0} = 1 \Rightarrow {\log _a}1 = 0$

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.