Question

How can we express \[\log 12\] in terms of \[\log 2\] and \[\log 3\]?

Answer 1

Hint: Here we will use the concept of logarithm and the properties of the logarithm. First, we will write the given logarithm term as a base of any number. Then we will apply the product rule for logarithms and express the given logarithm term as a product of 2 numbers. We will then simplify the numbers such that we get only \[\log 2\] and \[\log 3\]. We will then simplify the equation to get the required answer.

Complete step-by-step answer:
We have to express \[\log 12\] in terms of \[\log 2\] and \[\log 3\].
Let us assume that,
\[\log 12 = {\log _a}12\]….\[\left( 1 \right)\]
Where we let \[a\] as some base
We know the product rule of logarithm states that:
\[{\log _a}\left( {x \times y} \right) = {\log _a}x + {\log _a}y\]
Applying the above rule in equation \[\left( 1 \right)\], we get
\[\log 12 = \log \left( {4 \times 3} \right)\]
As we want to express our value in \[\log 2\] and \[\log 3\] term so will further simplify the value inside the bracket as,
\[ \Rightarrow \log 12 = \log \left( {2 \times 2 \times 3} \right)\]
Now separating the term we get,
\[ \Rightarrow \log 12 = \log 2 + \log 2 + \log 3\]
Adding the like terms, we get
\[ \Rightarrow \log 12 = 2\log 2 + \log 3\]

So we can express \[\log 12\] as \[2\log 2 + \log 3\].

Note:
The logarithm is defined as the power to which a certain number is raised to get another number. It is used to calculate the occurrence of the same factors in repeated multiplication. To denote the logarithm of \[x\] to base \[b\] we use \[{\log _b}\left( x \right)\]. We can find the value \[x\] by using the property \[{b^y} = x\] of the logarithm. The logarithm of the product of two values is equal to the sum of the logarithm of those values. The logarithm of the division of two values is equal to the difference of the logarithm of the two values with the same base. The logarithm function is the inverse of the exponential function.