Question

The value of x satisfying the logarithmic equation ${{\log }_{3}}\left( {{\log }_{2}}\left( {{\log }_{3}}x \right) \right)=1$ is
[a] x = 6561
[b] x = 65536
[c] x = 81
[d] x= 243

Answer 1

Hint: Assume ${{\log }_{2}}\left( {{\log }_{3}}x \right)=t$. Use the fact that if ${{\log }_{a}}\left( x \right)=y$, then $x={{a}^{y}}$. Hence find the value of t. Now assume ${{\log }_{3}}x=z$. Use the fact that if ${{\log }_{a}}\left( x \right)=y$, then $x={{a}^{y}}$. Hence find the value of z. Again use the fact that if ${{\log }_{a}}\left( x \right)=y$, then $x={{a}^{y}}$ and hence find the value of x. Verify your answer.

Complete step-by-step answer:
Let ${{\log }_{2}}\left( {{\log }_{3}}x \right)=t$.
Hence, we have
${{\log }_{3}}t=1$
We know that if ${{\log }_{a}}\left( x \right)=y$, then $x={{a}^{y}}$.
Using the above property, we get
$t={{3}^{1}}=3$
Hence, we have t = 3
Reverting to the original variable, we get
${{\log }_{2}}\left( {{\log }_{3}}x \right)=3$
Put ${{\log }_{3}}x=z$
Hence, we have
${{\log }_{2}}z=3$
We know that if ${{\log }_{a}}\left( x \right)=y$, then $x={{a}^{y}}$.
Using the above property, we get
$z={{2}^{3}}=8$
Hence, we have z=8
Reverting to the original variable, we get
${{\log }_{3}}x=8$
We know that if ${{\log }_{a}}\left( x \right)=y$, then $x={{a}^{y}}$.
Using the above property, we get
$x={{3}^{8}}=6561$
Hence, we have x = 6561.
Hence the solution of the given equation is x=6561
Hence, we conclude that option [a] is correct.

Note: [1] Verification:
We can verify the correctness of our solution by checking that the value of ${{\log }_{3}}\left( {{\log }_{2}}\left( {{\log }_{3}}x \right) \right)$ at x = 6561 is 1
We have
${{\log }_{3}}6561={{\log }_{3}}{{3}^{8}}$
We know that ${{\log }_{a}}{{a}^{n}}=n$
Hence, we have
${{\log }_{3}}6561=8$
Applying ${{\log }_{2}}$ on both sides of the equation, we get
${{\log }_{2}}\left( {{\log }_{3}}6561 \right)={{\log }_{2}}8={{\log }_{2}}{{2}^{3}}$
We know that ${{\log }_{a}}{{a}^{n}}=n$
Hence, we have
${{\log }_{2}}\left( {{\log }_{3}}6561 \right)=3$
Applying ${{\log }_{3}}$ on both sides of the equation, we get
${{\log }_{3}}\left( {{\log }_{2}}\left( {{\log }_{3}}6561 \right) \right)={{\log }_{3}}3=1$
Hence our solution is verified to be correct.
[2] A common mistake done by students is that they report solutions without verifying that the solutions are in the domain of the function or not. In solving questions involving inverse trigonometric functions and logarithmic functions, special care should be given to the domain of the functions involved. One can, however, avoid calculation of domain, just by verifying each solution at the end and discarding those solutions which do not satisfy the equation because they lead to values inside functions which are not in the domain of those functions.