Answer

Verified

431.4k+ views

**Hint:**First, we will use the formula to calculate the sum of the first term is \[a\] of a geometric progression is \[S = \dfrac{a}{{1 - a}}\] to simplify the power of given equation and then apply the \[2{\log _a}b = {\log _a}{b^2}\] in the obtained equation and then simplify to find the required value.

**Complete step-by-step answer:**We are given that \[{\left( {0.16} \right)^{{{\log }_{2.5}}\left\{ {\dfrac{1}{3} + \dfrac{1}{{{3^2}}} + ...} \right\}}}\].

Here, we will first use the formula to calculate the sum of the first term is \[a\]of a geometric progression is \[S = \dfrac{a}{{1 - a}}\] to simplify the power of above equation, we get

\[

\Rightarrow {\left( {0.16} \right)^{{{\log }_{2.5}}\left( {\dfrac{{\dfrac{1}{3}}}{{1 - \dfrac{1}{3}}}} \right)}} \\

\Rightarrow {\left( {0.16} \right)^{{{\log }_{2.5}}\left( {\dfrac{{\dfrac{1}{3}}}{{\dfrac{{3 - 1}}{3}}}} \right)}} \\

\Rightarrow {\left( {0.16} \right)^{{{\log }_{2.5}}\left( {\dfrac{{\dfrac{1}{3}}}{{\dfrac{2}{3}}}} \right)}} \\

\Rightarrow {\left( {0.16} \right)^{{{\log }_{2.5}}\left( {\dfrac{1}{2}} \right)}} \\

\Rightarrow {\left( {0.16} \right)^{{{\log }_{2.5}}\left( {0.5} \right)}} \\

\Rightarrow {\left( {0.4} \right)^{2{{\log }_{2.5}}\left( {0.5} \right)}} \\

\]

Applying the log rule,\[2{\log _a}b = {\log _a}{b^2}\] in the above equation and simplify, we get

\[

\Rightarrow {\left( {0.4} \right)^{{{\log }_{2.5}}{{\left( {0.5} \right)}^2}}} \\

\Rightarrow {\left( {0.4} \right)^{{{\log }_{2.5}}\left( {0.25} \right)}} \\

\]

Using the logarithm value,\[{\log _{2.5}}\left( {0.25} \right) = - 1.51294...\] in the above equation, we get

\[

\Rightarrow {\left( {0.4} \right)^{ - 1.5124...}} \\

\Rightarrow 4 \\

\]

**So, the required value is 4.**

**Note:**The key point here is to use the properties of the logarithm and the trigonometric rule right in the question or else it will be really confusing to solve. The power rule can be used for fast exponent calculation using multiplication operation. Students should make use of the appropriate formula of logarithms wherever needed and solve the problem. In mathematics, if the base value in the logarithm function is not written, then the base is \[e\].

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

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

Trending doubts

Difference Between Plant Cell and Animal Cell

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

Give 10 examples for herbs , shrubs , climbers , creepers

10 examples of evaporation in daily life with explanations

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

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

Name 10 Living and Non living things class 9 biology CBSE