Question

The domain of the function $f(x) = {\log _2}[{\log _3}({\log _4}x)]$ is

1. $( - \infty ,4)$
2. $(4,\infty )$
3. $(0,4)$
4. $(1,\infty )$
5. $( - \infty ,1)$

Answer 1

The domain of a function is the set of all possible inputs for the function . The logarithm is the inverse function to exponentiation. To solve this problem you should use log property .
As, $\log x > 0$ (where x is always positive )

Complete step-by-step solution:
Function - ${\log _2}[{\log _3}({\log _4}x)]$
${\log _4}x$; $x > 0$
$x$ should be always positive
Let ${\log _4}x$ be t
${\log _3}t$; $t > 0$
As, $t > 0$ where t is ${\log _4}x$
So, ${\log _4}x > 0$
${\log _4}x > 0$
The base 4 of log will shift towards right side
$x > {4^0}$
$x > 1$
${\log _2}[{\log _3}({\log _4}x)]$
Let ${\log _3}({\log _4}x)$ be m
${\log _2}m$ where $m > 0$
As, $m > 0$ where m is ${\log _3}({\log _4}x)$
So ${\log _3}({\log _4}x) > 0$
${\log _3}({\log _4}x) > 0$
The base 3 of log will shift towards right side
${\log _4}x > {3^0}$
${\log _4}x > 1$
The base 4 of log will shift towards right side
$x > {4^1}$
$x > 4$
Intersection of all the values of x s
Therefore the domain of the function is
$x > 0$ , $x > 1$ , $x > 4$
Domain - $x \in (4,\infty )$
Option (2) is correct

Note: The range of function is the set of output of the function achieved when it is applied to its whole set of the output. As we know in $\log x$ $x > 0$. and the x is always positive . First we compared ${\log _4}x$ from zero then which gave $x > 1$ then we compared ${\log _3}({\log _4}x)$ from zero which gave $x > 4$.