Question

How do you express $\log 36$ in terms of $$\log 2$$ and $$\log 3$$?

Answer 1

Hint: We use logarithms so as to express the exponential part of a number. It=n the above question we need to reduce the term log36 into log2 and log3 by using logarithmic rules.2 and 3 should be written in the power of 36 in such a way that it should be the product of 36.

Complete step by step answer:
The question states to write log36 in terms of log2 and log3 and this can be done by using logarithmic rules which are given as follows:
$$\log a^b=b.\log a$$
$$\log (a.b)=\log a+\log b$$
So, now the first step will be splitting the number 36 into products of the above numbers 2 and 3. As factors of $36=6\times 6$ .It can be further written as
$$\begin{gathered} 36=6^2 \\ 36=(3\cdot 2{)}^2 \end{gathered}$$
Now, applying the above given first property we get the expression as follows:
$$\log (2\cdot 3{)}^2=2\cdot \log (2\cdot 3)$$
The term $$\log (2\cdot 3)$$ can further split by using the second property mentioned above.
$$2\log (2\cdot 3)=2.(\log 2+\log 3)$$
So, it can be finally written as:
$$\log 36=2.(\log 2+\log 3)$$
Therefore, the above expression is written in terms of log2 and log3.

Note:
In the above question, log36 base will be 6 and it can be written as $${\log }_{6}36$$.The reason for base being 6 is because $$36=6^2$$ and as it is clearly given in the equation that 6 is the base and power is 2, so by applying on log on both sides we get $$2={\log }_{6}36$$.
Additional information: An important thing to know that the expression $${\log }_{a}a$$ is log a with base a is equal to 1. This is because $${\log }_{a}a=1$$ can be written as $$a=a^1$$.Here, a has the power of 1, which is
equal to a.