Answer

Verified

365.1k+ views

**Hint**: Here in this question, we have to find the correct option for the given question. First consider a right triangle with hypotenuse side ‘c’ then ‘a’ and ‘b’ are the other sides of the triangle and next take a Pythagoras theorem i.e., \[op{p^2} + ad{j^2} = hy{p^2}\] for the given right triangle on applying a logarithm with base \[{c^2}\] to the equation of Pythagoras and further simplify by properties of logarithms we get the required solution.

**Complete step-by-step answer**:

Given, the right triangle with sides of a, b and c where ‘c’ is the largest of the three numbers. In the right triangle the largest side should be a hypotenuse, so ‘c’ be the hypotenuse side.

Now Pythagoras theorem which stated as “In a right- angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides” i.e., \[op{p^2} + ad{j^2} = hy{p^2}\]

For a given triangle

\[ \Rightarrow {a^2} + {b^2} = {c^2}\]-----(1)

Apply a log on both side with base \[{c^2}\], then we have

\[ \Rightarrow {\log _{{c^2}}}\left( {{a^2} + {b^2}} \right) = {\log _{{c^2}}}\left( {{c^2}} \right)\] -----(2)

As we know the logarithm function with base \[e\] become \[{\log _e}e = 1\], then RHS of equation (2) becomes

\[ \Rightarrow {\log _{{c^2}}}\left( {{a^2} + {b^2}} \right) = 1\]-------(3)

Now, apply the change-of-base formula for LHS in equation (3), then

\[ \Rightarrow \dfrac{{{{\log }_c}\left( {{a^2} + {b^2}} \right)}}{{{{\log }_c}{c^2}}} = 1\]

Now, apply a power rule of logarithm i.e., \[\log \left( {{m^n}} \right) = n\log m\], in denominator of LHS, then we have

\[ \Rightarrow \dfrac{{{{\log }_c}\left( {{a^2} + {b^2}} \right)}}{{2{{\log }_c}c}} = 1\]

Or

\[ \Rightarrow \dfrac{1}{2}\left( {\dfrac{{{{\log }_c}\left( {{a^2} + {b^2}} \right)}}{{{{\log }_c}c}}} \right) = 1\]

Again, apply the property \[{\log _e}e = 1\] in denominator of LHS, we get

\[ \Rightarrow \dfrac{1}{2}{\log _c}\left( {{a^2} + {b^2}} \right) = 1\]

Multiply 2 on both side

\[ \Rightarrow {\log _c}\left( {{a^2} + {b^2}} \right) = 2\]

It’s a required solution.

Therefore, option (C) is correct.

**So, the correct answer is “Option C”.**

**Note**: The logarithmic function is a reciprocal or the inverse of exponential function. To solve the question, we must know about the properties of the logarithmic function. There are properties on addition, subtraction, product, division etc., on the logarithmic functions. We have to change the base of the log function and to simplify the given question.

Recently Updated Pages

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

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

Advantages and disadvantages of science

Trending doubts

Which are the Top 10 Largest Countries of the World?

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

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

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

10 examples of evaporation in daily life with explanations