Question

The value of \[{\text{lo}}{{\text{g}}_{{\text{10}}}}{\text{0}}{\text{.001}}\]is equal to
A. −3
B. 3
C. −2
D. 2

Answer 1

Hint: We can convert the decimal into exponential form. Then we can find a log as the power of 10 in the exponential form. We know that \[{\log _a}{a^b} = b\]. Applying the value of a as 10 and b as the power of 10 in the above equation, we get the required log value.

Complete step by step Answer:

We need to find the logarithm of a decimal number. We can convert the given decimal to its standard exponential form.
Writing ${\text{0}}{\text{.001}}$in exponential form, we get,
${\text{0}}{\text{.001 = 1}}{{\text{0}}^{{\text{ - 3}}}}$
Now we can take the logarithm, ${\text{lo}}{{\text{g}}_{{\text{10}}}}$ of a number is the power of 10 when it is written in its exponential form. We get,
${\log _{10}}0.001 = {\log _{10}}{10^{ - 3}}$
We know that \[{\log _a}{a^b} = b\]. So, \[{\log _{10}}{10^b} = b\] and \[{\log _{10}}{10^1} = 1\]
$ \Rightarrow {\log _{10}}{10^{ - 3}} = - 3$
So, the value of \[{\text{lo}}{{\text{g}}_{{\text{10}}}}{\text{0}}{\text{.001}}\] is equal to -3.
Therefore, the correct answer is option A.

Note: The concept of logarithm is used in this problem. The logarithm of any decimal can be found out by converting it into its exponential form. The log values of the decimal part can be found from a logarithm table. ${\text{lo}}{{\text{g}}_{{\text{10}}}}$ is the log to the base 10. Logarithm to the base e is known as a natural log and it is represented by $\ln $. It is the power of e when the decimal is written as the power of e. The inverse operation of the log is antilog. For the natural logarithm, the inverse operation is the exponential function.