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1) ${\log _m}n = \dfrac{{{{\log }_a}n}}{{{{\log }_a}m}}$

2) $\log {m^n} = n\log m$

3) ${\log _a}a = 1$

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$