Question

The value of \[\log 0.0001\] to the base $0.1$ is
A) $ - 4$
B) $4$
C) $1$
B) $-2$

Answer 1

Hint: Use the definition of logarithm, to find a relationship between the required logarithmic value, and the given numbers 0.001 and 0.1. We can then solve it to get the required value. Alternatively, we can use the change of base rule and represent it in terms of base 10.

Complete step by step solution:
We are asked to find the logarithmic value of $0.0001$ to the base $0.1$. In mathematics, logarithm is the inverse function to exponentiation. That means that the logarithm of a given number $a$ to which another fixed number, the base b, must be raised, so as to produce that number $a$. We can write this as ${\log _b}a = x$ implies that ${b^x} = a$. We will use the definition to find the given logarithm in the question which is ${\log _{0.1}}0.0001$.
Let us assume the value of ${\log _{0.1}}0.0001$ to be $x$.
Then by definition we have ${0.1^x} = 0.0001$. - - - - - - - - - - - - - - - - - - - - - - (1)
Now, note that $0.1$ can be written as $\dfrac{1}{{10}}$ which is equal to ${10^{ - 1}}$. Again, we can see that $0.0001$ can be written as $\dfrac{1}{{10000}} = \dfrac{1}{{{{10}^4}}}$ which is equal to ${10^{ - 4}}$.
So now we will use these values in equation (1) to get, ${\left( {{{10}^{ - 1}}} \right)^x} = {10^{ - 4}}$.
Using the power rule of exponentiation ${\left( {{a^m}} \right)^n} = {a^{m \times n}}$, we get ${10^{ - x}} = {10^{ - 4}}$.
Now since bases are both equal to 10 on both sides of the equation the powers should be equal.
$ \Rightarrow - x = - 4$
$ \Rightarrow x = 4$
Therefore, the value of \[\log 0.0001\] to the base $0.1$ is $4$. Hence, the correct option is (B) $4$.

Note:
Alternative method of solving the question involves using the change of base rule for logarithms which states that ${\log _b}a = \dfrac{{{{\log }_c}a}}{{{{\log }_c}b}}$ to write given logarithm in terms of logarithms in base 10, whose values are relatively easier to find. The method used in the solution for this question is not always applicable. So, one has to proceed with the relatively easier change of base rule method.