Question

Expand the following logarithmic expression:
${{\log }_{10}}385$

Answer 1

Hint: We are asked to find the expansion of ${{\log }_{10}}385$. To expand this first of all we are going to write the prime factorization of 385. After writing the prime factorization of 985 we will apply the property of logarithm which says that ${{\log }_{10}}\left( a\times b \right)={{\log }_{10}}a+{{\log }_{10}}b$. We are going to use this property because we have the factors of 385 in multiplication form so simplification of that term will be easy using this logarithmic property.

Complete step by step answer:
We have to expand the following expression:
${{\log }_{10}}385$
First of all, we are going to write the prime factorization of 385. In the below, we have written the prime factorization of 385.
$385=5\times 7\times 11\times 1$
Now, substituting the above prime factorization of 385 in ${{\log }_{10}}385$ we get,
${{\log }_{10}}\left( 5\times 7\times 11\times 1 \right)$
The above logarithm is in the form of ${{\log }_{10}}\left( a\times b\times c\times d \right)$ and there is a property of logarithm in which:
${{\log }_{10}}\left( a\times b\times c\times d \right)={{\log }_{10}}a+{{\log }_{10}}b+{{\log }_{10}}c+{{\log }_{10}}d$
So, we are going to use the above property of logarithm in simplifying this logarithmic expression ${{\log }_{10}}\left( 5\times 7\times 11\times 1 \right)$ as follows:
${{\log }_{10}}\left( 5\times 7\times 11\times 1 \right)={{\log }_{10}}5+{{\log }_{10}}7+{{\log }_{10}}11+{{\log }_{10}}1$………… Eq. (1)
We know the values of ${{\log }_{10}}5,{{\log }_{10}}7,{{\log }_{10}}11,{{\log }_{10}}1$ as follows:
$\begin{align}
  & {{\log }_{10}}5=0.698 \\
 & {{\log }_{10}}7=0.845 \\
 & {{\log }_{10}}11=1.041 \\
 & {{\log }_{10}}1=0 \\
\end{align}$
Using the above logarithmic values in simplifying eq. (1) we get,
$\begin{align}
  & {{\log }_{10}}\left( 5\times 7\times 11\times 1 \right)=0.698+0.845+1.041+0 \\
 & \Rightarrow {{\log }_{10}}\left( 5\times 7\times 11\times 1 \right)=2.584 \\
\end{align}$
From the above solution, we have expanded the logarithmic expression ${{\log }_{10}}385$ given in the above problem to 2.584.
Hence, the value of ${{\log }_{10}}385$ is equal to 2.584.

Note:
 You might have thought that how one should know to approach the above problem in the manner that we have stated above.
The answer is we have to expand the logarithmic expression i.e. ${{\log }_{10}}385$. Now, if we have given the logarithmic table then we can find its value but we have not provided any logarithmic table so we try to factorize 385 after that you will find the prime factors are multiplied to each other. And then it can strike that there is a property of logarithm in which ${{\log }_{10}}\left( a\times b\times c\times d \right)={{\log }_{10}}a+{{\log }_{10}}b+{{\log }_{10}}c+{{\log }_{10}}d$ and then we put the respective values of the logarithm.
A point to be noted in this problem is that it is always better to remember the values of log base 10 $\left( {{\log }_{10}} \right)$ with arguments from 1 to 12. It will save your time in solving questions in competitive examinations.