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

Answer

Verified

181.5k+ views

**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.

Recently Updated Pages

Are these pairs of triangles congruent In each case class 9 maths CBSE

To construct a convex quadrilateral which of the following class 9 maths CBSE

Which of the following expressions are polynomials class 9 maths CBSE

Which number added to the polynomial 3x2 4x 1 gives class 9 maths CBSE

areadfrac12times base times height Which of the following class 9 maths CBSE

Draw an isosceles triangle with each of equal sides class 9 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

A Short Paragraph on our Country India