Question

How do you solve $ \log \left( \log x \right)=1 $ ?

Answer 1

Hint: We first decide on the base of the logarithmic base. The base of the log in general cases is 10. Then we use the logarithmic identity of $ {{\log }_{a}}x=y\Rightarrow {{a}^{y}}=x $ . We assume the value of $ {{\log }_{10}}x=m $ . We solve the logarithmic equation of m first. Then we apply the same identity on the value of m again for $ {{\log }_{10}}x=10 $. From the equation, we solve the value of x.

Complete step by step answer:
We need to solve the logarithmic equation $ \log \left( \log x \right)=1 $ .
We know that if the base in case of logarithm equation is not mentioned then we take the base as 10.
Therefore, the equation $ \log \left( \log x \right)=1 $ becomes $ {{\log }_{10}}\left( {{\log }_{10}}x \right)=1 $ .
We also have the logarithmic identity $ {{\log }_{a}}x=y\Rightarrow {{a}^{y}}=x $ .
Let us assume $ {{\log }_{10}}x=m $ . This gives $ {{\log }_{10}}m=1 $ .
We apply the theorem $ {{\log }_{a}}x=y\Rightarrow {{a}^{y}}=x $ to get $ m={{10}^{1}}=10 $ .
We know $ {{\log }_{10}}x=m $ . We replace the value of m in $ {{\log }_{10}}x=m $ .
We get $ {{\log }_{10}}x=10 $ .
We again apply the theorem of $ {{\log }_{a}}x=y\Rightarrow {{a}^{y}}=x $ and get
 $ \begin{align}
  & {{\log }_{10}}x=10 \\
 & \Rightarrow x={{10}^{10}} \\
\end{align} $
Therefore, the solution of the equation $ \log \left( \log x \right)=1 $ is $ x={{10}^{10}} $ .

Note:
The logarithm is used to convert a large or very small number into an understandable domain. For the theorem to work the usual conditions of logarithm will have to follow. For $ {{\log }_{a}}x=y $ , the value of x has to be greater than 0. The logarithm on both sides of the equation $ x={{10}^{10}} $ gives the equation $ \log \left( \log x \right)=1 $ in return.