Question

How do you expand $\log \left( {10,000x} \right)$ ?

Answer 1

Hint:The given problem deals with the use of logarithms. It focuses on the basic definition of the logarithm function and its properties. For such type of questions that require us to simplify logarithmic expressions, we need to have knowledge of all the properties of logarithmic function and applications of each one of them.

Complete step by step answer:
In the given problem, we are required to expand $\log \left( {10,000x} \right)$. This simplification and expansion can be done with the help of logarithmic properties.
So, we have, $\log \left( {10,000x} \right)$
Using the logarithmic property ${\log _z}(x \times y) = {\log _z}x + {\log _z}y$, we get,
$ \Rightarrow \log \left( {10,000x} \right) = \log \left( {10000 \times x} \right)$
$ \Rightarrow \log \left( {10000x} \right) = \log \left( {10000} \right) + \log x$
We know that $10000 = {10^4}$. So, we get,
$ \Rightarrow \log \left( {10000x} \right) = \log \left( {{{10}^4}} \right) + \log x$
Now, by basic definition of logarithmic function and understanding of interconversion of logarithmic function to exponential function, we know that ${\log _{10}}(10) = 1$. Similarly, ${\log _{10}}({10^4}) = 4$.
$ \Rightarrow \log \left( {10000x} \right) = 4 + \log x$

Hence, $\log \left( {10,000x} \right)$ can be expanded as $4 + \log x$ by the use of logarithmic properties and identities.

Note: The given problem involves use of properties and identities of log function and hence requires us to have a thorough knowledge of the same. We also need to have a basic idea about the applications of the identities and properties in such questions.