Question

How do you evaluate ${{\log }_{6}}36$?

Answer 1

Hint: First we will write 36 as ${{6}^{2}}$. Then we will use the property of logarithm that $\log {{m}^{n}}=n\log m$ to simplify the expression then we will use the property ${{\log }_{a}}a=1$ and apply it to the obtained equation to get the desired answer.

Complete step-by-step solution:
We have been given an expression ${{\log }_{6}}36$.
We have to find the value of the given expression.
We know that base e and base 10 are common bases used to represent the logarithm. A logarithm with base 10 is common logarithm and natural logarithm is different.
Now, we can rewrite the given expression as
$\Rightarrow {{\log }_{6}}{{6}^{2}}$ because we know that $36=6\times 6={{6}^{2}}$ .
Now, we know that by logarithm property we have $\log {{m}^{n}}=n\log m$.
Now, applying the property to the above obtained equation we will get
$\Rightarrow 2{{\log }_{6}}6$
Now, we know that ${{\log }_{a}}a=1$.
Now, substituting the value to the above obtained equation we will get
$\begin{align}
  & \Rightarrow 2\times 1 \\
 & \Rightarrow 2 \\
\end{align}$
So, on simplifying the given expression ${{\log }_{6}}36$ we get the value $2$.

Note: We know that logarithm is the special form of exponentiation. Alternatively we can solve the given expression by using exponential rule and using the definition of a logarithm. We know that ${{\log }_{a}}x=b$ is equal to the ${{a}^{b}}=x$ .
So when we compare the given expression with the above explained property we will get
$\Rightarrow {{\log }_{6}}36=b$
Therefore we can write it as
$\Rightarrow {{6}^{b}}=36$
Now, we know that $36=6\times 6={{6}^{2}}$
So, substituting the value we will get
$\Rightarrow {{6}^{b}}={{6}^{2}}$
On comparing the LHS and RHS we will get
$b=2$
So we get the value $\Rightarrow {{\log }_{6}}36=2$