Question

Given the expression: $\log 2 = 0.3010$ and $\log 3 = 0.4771$, find the value of $\log 12$.

Answer 1

Hint: In order to find the value of $\log 12$, we need to know about the rules of logarithms that are being followed. We need rules at each step of solving the equations of this question. For example, we know that $\log \left( {m.n} \right)$ can be expanded and written as $\log \left( {m.n} \right) = \log m + \log n$.
Formula used:
$\log \left( {m.n} \right) = \log m + \log n$
$\log {b^a} = a\log b$

Complete step-by-step solution:
We are given with some piece of information such as:
$\log 2 = 0.3010$ and $\log 3 = 0.4771$
We need to find the value of $\log 12$ using the information given.
For that we need to expand the value of $\log 12$, so we can expand the value of 12 as it’s factors as $12 = 3 \times 4$.
So, replacing 12 in $\log 12$ with the above value, we can write it as:
$\log 12 = \log \left( {3 \times 4} \right)$
From the laws or rules of logarithms, we know that $\log \left( {m.n} \right)$ can be written as $\log \left( {m.n} \right) = \log m + \log n$.
So, using this law, comparing $\log \left( {3 \times 4} \right)$ with $\log \left( {m.n} \right)$, we get:
$m = 3$ and $n = 4$
Substituting the value of m and n in the equation $\log \left( {m.n} \right) = \log m + \log n$, we get:
$\log \left( {3 \times 4} \right) = \log 3 + \log 4$
Since, we know that 4 can be written as $4 = {2^2}$, substituting this in the above equation:
$\log \left( {3 \times 4} \right) = \log 3 + \log {2^2}$ …..(1)
From the rules of logarithms, we know that we can write $\log {a^b}$ as $b\log a$.
So, by comparing $\log {a^b}$ with $\log {2^2}$, we can write it as $2\log 2$.
Substituting this value in the equation 1, we get:
$\log \left( {3 \times 4} \right) = \log 3 + 2\log 2$
Now, substituting the value of $\log 2 = 0.3010$ and $\log 3 = 0.4771$ in the above equation, we get:
$ \Rightarrow \log \left( {3 \times 4} \right) = 0.4771 + 2\left( {0.3010} \right)$
On solving, we get:
$ \Rightarrow \log \left( {3 \times 4} \right) = 0.4771 + 0.6020$
$ \Rightarrow \log \left( {3 \times 4} \right) = 1.0791$
$ \Rightarrow \log 12 = 1.0791$
Therefore, the value of $\log 12$ is $1.0791$.

Note: Remember, $\log m.\log n \ne \log m + \log n$ and $\dfrac{{\log m}}{{\log n}} \ne \log m - \log n$ as because the rules for the logarithms are as: $\log \left( {\dfrac{m}{n}} \right) = \log m - \log n$ and $\log \left( {m.n} \right) = \log m + \log n$, so do not get confused in the values. Since, we know that 12 can also be expanded as $12 = 6 \times 2$, but we expanded it as $12 = 3 \times 4$ because we were given the logarithmic values of these two. So, always expand according to the details given in the question.