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

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,

& 2\left| \\!{\underline {\, 60 \,}} \right. \\\ & 2\left| \\!{\underline {\, 30 \,}} \right. \\\ & 3\left| \\!{\underline {\, 15 \,}} \right. \\\ & 5\left| \\!{\underline {\, 5 \,}} \right. \\\ & 1 \\\ \end{aligned}$$ Therefore, the factors of 60 are $$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.$$