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What is the value of ${\log _3}{\log _2}{\log _{\sqrt 5 }}\left( {{5^4}} \right)$?

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Hint: In this question, we are given an expression in logarithmic terms and we have been asked to find its value. Start by simplifying the rightmost term. Use the base conversion formula to simplify the logarithmic term containing 5. Then using its powers, simplify it further. Use the 2nd and 3rd property mentioned below to further simplify the equation and find the most simplified answer independent of log.

Formula used:
1) ${\log _m}n = \dfrac{{{{\log }_a}n}}{{{{\log }_a}m}}$
2) $\log {m^n} = n\log m$
3) ${\log _a}a = 1$

Complete step-by-step solution:
We are given an expression in log and we have been asked to find its value. We will use various logarithmic properties to find the required value.
$ \Rightarrow {\log _3}{\log _2}{\log _{\sqrt 5 }}\left( {{5^4}} \right)$ …. (given)
We can write $\sqrt 5 = {5^{\dfrac{1}{2}}}$, Putting this in the above equation,
$ \Rightarrow {\log _3}{\log _2}{\log _{{5^{\dfrac{1}{2}}}}}\left( {{5^4}} \right)$…. (1)
Now, we will use base conversion formula to simplify the equation.
Formula - ${\log _m}n = \dfrac{{{{\log }_a}n}}{{{{\log }_a}m}}$. Using this in the above equation,
$ \Rightarrow {\log _{{5^{\dfrac{1}{2}}}}}{5^4} = \dfrac{{\log {5^4}}}{{\log {5^{\dfrac{1}{2}}}}}$
Using property $\log {m^n} = n\log m$ to simplify the above equation,
$ \Rightarrow {\log _{{5^{\dfrac{1}{2}}}}}{5^4} = \dfrac{{4\log 5}}{{\dfrac{1}{2}\log 5}}$
Simplifying the equation,
$ \Rightarrow {\log _{{5^{\dfrac{1}{2}}}}}{5^4} = 4 \times 2$
$ \Rightarrow {\log _{{5^{\dfrac{1}{2}}}}}{5^4} = 8$
Putting this in equation (1),
$ \Rightarrow {\log _3}{\log _2}8$
We can write $8 = {2^3}$ in the above equation.
$ \Rightarrow {\log _3}{\log _2}{2^3}$
Using property $\log {m^n} = n\log m$ again.
$ \Rightarrow {\log _3}3{\log _2}2$
There is a property which says that ${\log _a}a = 1$
Therefore, ${\log _2}2 = 1$
It gives us –
$ \Rightarrow {\log _3}3$
Using the same property ${\log _a}a = 1$ again.
$ \Rightarrow {\log _3}3 = 1$

Hence, ${\log _3}{\log _2}{\log _{\sqrt 5 }}\left( {{5^4}} \right) = 1$

Note: We have in mind that, while simplifying ${\log _{{5^{\dfrac{1}{2}}}}}{5^4}$, we used two different properties. Instead of that, we can use one single property - ${\log _{{a^b}}}{m^n} = \dfrac{n}{b} \times \dfrac{{\log m}}{{\log a}}$. This will give the same answer but it will reduce the number of steps. You can either use the 2 properties as we have used or you can use one single property mentioned here.