Question

Find the value of x if ${\log }_{10}{\log }_{10}{\log }_{10}x=0$

Answer 1

Hint: use the basic definition of logarithm that is if we have $a^x=N$, then by taking log on both sides, we get ${\log }_{a}N=x$. And vice versa is also true. Use this to solve the given problem

We have equation given;
${\log }_{10}{\log }_{10}{\log }_{10}x=0$………………. (1)
Here, we need to apply the basic definition of logarithm function i.e. if we have the expression $a^x=N$, we can convert it into log by taking log to both sides with base ‘a’.
Now, taking log to both sides with base ‘a’, we get;
${\log }_a{a}^x={\log }_{a}N$…………………. (2)
Now, we know identity of logarithm function as;
${\log }_c{m}^n=n{\log }_{c}m$
Rewriting the equation (2), with the help of above equation, we get;
$x{\log }_{a}a={\log }_{a}N$
As we know ${\log }_{c}C=1$ i.e. value of any log on the same base is 1.
Hence, we get;
$x={\log }_{a}N$
Therefore, if $a^x=N$, then we can write this equation in logarithmic form as;
$x={\log }_{a}N$and vice – versa is also true i.e. if $x={\log }_{a}N$then $a^x=N$.
Now, using the given property with equation (1) we get
Since,
$\begin{array}{rl}& {\log }_{10}{\log }_{10}{\log }_{10}x=0\\ & {\log }_{10}{\log }_{10}x=10{}^{\circ }\end{array}$
Now, we know that $a{}^{\circ }=1,$ therefore; above equation can be written as;
${\log }_{10}{\log }_{10}x=1$………………… (3)
Now, we can use same property as explained initially in the solution i.e. if ${\log }_{a}N=x$, then $a^x=N$
Using the same property with equation (3), we get,
Since, we have;
$\begin{array}{rl}& {\log }_{10}{\log }_{10}x=1\\ & {\log }_{10}x={10}^1=10\end{array}$
Hence,
${\log }_{10}x=10$……………… (4)
Now, we can use the same property again with equation (4), we get
Since,
$\begin{array}{rl}& {\log }_{10}x=10\\ & x={10}^{10}\end{array}$
Hence, on solving ${\log }_{10}{\log }_{10}{\log }_{10}x=0$, we get $x={10}^{10}$

Note: One can go wrong while doing conversion of ${\log }_{a}N=x$to $N=a^x$. One can write $a=N^x\text{ or }N=x^a$ which is wrong. One can give answer x = 1, as log 1= 0, which is wrong, because this will be true for ${\log }_{10}x=0$ but we equation as ${\log }_{10}{\log }_{10}{\log }_{10}x=0$. Hence x = 1 will not be the correct solution to the given equation.