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
\[
  36 = {6^2} \\
  36 = {(3 \cdot 2)^2} \\
 \]
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.