Question

Expand the following:
log 1000

Answer 1

Hint: We will be using the concepts of logarithmic function to solve the problem. We will first write 1000 as an exponent of 10 then we will use the logarithmic identities to further simplify the problem.

Complete step-by-step answer:
Now, we have to expand log 1000. So, now we will first express 1000 as ${{10}^{n}}$ where n is any integer. So, we have,
$\begin{align}
  & 1000={{10}^{n}} \\
 &\Rightarrow (10\times 10\times 10 )={{10}^{n}} \\
 &\Rightarrow {{10}^{3}}={{10}^{n}} \\
 & \Rightarrow n=3 \\
\end{align}$
Therefore, we have $1000={{10}^{3}}$.
Now, we know the logarithmic identities that,
${{\log }_{a}}{{b}^{n}}=n{{\log }_{a}}b$
So, now we have,
$\begin{align}
  & \log 1000=\log {{10}^{3}} \\
 & =3\log 10 \\
\end{align}$
So, log 1000 can be expanded as $3\log 10$.

Note: To solve these type of questions it is important to note that we have to used logarithmic identity that ${{\log }_{a}}{{b}^{n}}=n{{\log }_{a}}b$. Also, it should be noted that 1000 can be written as ${{10}^{3}}$, so the problem gets simplified. Also, there is an alternate method:
$\begin{align}
  & \log 1000=\log (10\times 10\times 10 )\\
 & =\log 10+\log 10+\log 10 \\
 & =3\log 10 \\
\end{align}$
We have used the identity that $\log ab=\log a+\log b$.