Question

If $A = {\log _2}{\log _2}{\log _4}256 + 2{\log _{\sqrt 2 }}2$, then find $A$.
A) $2$
B) $3$
C) $5$
D) $7$

Answer 1

Hint:
Here we have the sum of two logarithmic terms. So we find the logarithms separately. In the first term log is applied continuously. There we have to start from the last and come in reverse.
Useful formula:
Logarithmic function to a certain base is defined as
$x = {b^n} \Rightarrow {\log _b}x = n$, where both $b$ and $x$ are positive.

Complete step by step solution:
Given $A = {\log _2}{\log _2}{\log _4}256 + 2{\log _{\sqrt 2 }}2$.
We are asked to find $A$.
Since here is a series of log functions we can start from the last one.
First we have to find ${\log _4}256$.
$x = {b^n} \Rightarrow {\log _b}x = n$, where both $b$ and $x$ are positive.
In the condition let $x = 256,b = 4$
$256 = {4^n} \Rightarrow {\log _4}256 = n$
We have ${4^4} = 4 \times 4 \times 4 \times 4 = 256$.
So we get $n = 4$. Substituting this we have,
$256 = {4^4} \Rightarrow {\log _4}256 = 4$
Substituting the value of ${\log _4}256$, the given expression is reduced to $A = {\log _2}{\log _2}4 + 2{\log _{\sqrt 2 }}2$
Now let us find the value of ${\log _2}4$.
$x = {b^n} \Rightarrow {\log _b}x = n$, where both $b$ and $x$ are positive.
In the condition let $x = 4,b = 2$
$4 = {2^n} \Rightarrow {\log _2}4 = n$
We have ${2^2} = 4$. This gives $n = 2$.
Substituting the value we get,
$ \Rightarrow {\log _2}4 = 2$
Using this again our expression is reduced to $A = {\log _2}2 + 2{\log _{\sqrt 2 }}2$.
Now let us find ${\log _2}2$
By the same method used above,
we have ${2^1} = 2 \Rightarrow {\log _2}2 = 1$
Therefore, we have $A = 1 + 2{\log _{\sqrt 2 }}2$
Now it remains to find ${\log _{\sqrt 2 }}2$.
We have ${(\sqrt 2 )^2} = 2 \Rightarrow {\log _{\sqrt 2 }}2 = 2$
So $A = 1 + 2{\log _{\sqrt 2 }}2 = 1 + (2 \times 2)$
$ \Rightarrow A = 1 + 4 = 5$

$\therefore $ The answer is option C.

Additional information:
Logarithmic functions are the inverses of exponential functions. The domain of logarithmic function is real numbers greater than zero and the range is real numbers.

Note:
Here the bases are all specified in logarithmic functions. If a log has no base written, we should generally assume that the base is \[10\]. The other important log is the “natural” or base-e log, denoted as $\ln (x)$.