Question

If we have the logarithm expression \[{{\log }_{10}}2=a\] and \[{{\log }_{10}}3=b,\] then log 60 can be expressed in terms of a and b as
\[\left( a \right)a+b+1\]
\[\left( b \right)a+b-1\]
\[\left( c \right)a-b+1\]
\[\left( d \right)a-b-1\]

Answer 1

Hint: First of all, we will write all the factors of the numbers 60 and then try to find only those factors of 60 which has 2 and 3 in it. Then by using the fact that \[60=6\times 10\] and \[{{\log }_{10}}10=1\] then again splitting log 6 as \[6=2\times 3\] and using \[\log \left( mn \right)=\log m+\log n\] we will get the required result.

Complete step-by-step solution:
The factors of 60 can be written by taking the LCM of 60. The LCM of 60 is given as,
\[\begin{align}
  & 2\left| \!{\underline {\,
  60 \,}} \right. \\
 & 2\left| \!{\underline {\,
  30 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & 1 \\
\end{align}\]
Therefore, the factors of 60 are
\[60=2\times 2\times 3\times 5\]
So, the factor of 60 can be written as
\[60=\left( 5\times 2 \right)\times \left( 3\times 2 \right)\]
\[\Rightarrow 60=10\times 6\]
So, 60 can be written as \[10\times 6.\]
We have to find the value of log 60.
\[\Rightarrow {{\log }_{10}}60\]
\[\Rightarrow {{\log }_{10}}\left( 6\times 10 \right)\]
Now, using the log property, we have, \[\log \left( mn \right)=\log m+\log n.\]
By substituting m = 6 and n = 10, we get,
\[\Rightarrow {{\log }_{10}}60={{\log }_{10}}6+{{\log }_{10}}10\]
Now, \[{{\log }_{10}}10=1\]
\[\Rightarrow {{\log }_{10}}60={{\log }_{10}}6+1\]
Now, again we will split 6 as 2.3 and use \[\log \left( mn \right)=\log m+\log n.\] We will put m = 3 and n = 2. So, we get,
\[\Rightarrow {{\log }_{10}}6={{\log }_{10}}3+{{\log }_{10}}2\]
\[\Rightarrow {{\log }_{10}}60={{\log }_{10}}3+{{\log }_{10}}2+1\]
Now, \[{{\log }_{10}}3=b\] is given in the question and \[{{\log }_{10}}2=a\] is also given in the question. Substituting these values in the above equation, we get,
\[\Rightarrow {{\log }_{10}}60=a+b+1\]
So, the value of \[{{\log }_{10}}60\] in terms of a and b is a + b + 1.
Hence, the correct option is (a).

Note: We should note that 60 was split as \[60=2\times 2\times 3\times 5\] and we did not use this ‘5’ here while calculating \[{{\log }_{10}}60.\] This is so as the value of \[{{\log }_{10}}5\] was not given and it can be determined through but we needed our answer in the form of a and b. So, we did not use ‘5’, instead, we used \[60=6\times 10.\] So, that we can split 6 as \[2\times 3.\]