Question

Find the value of the expression \[7\log \left( {\dfrac{{16}}{{15}}} \right) + 5\log \left( {\dfrac{{25}}{{24}}} \right) + 3\log \left( {\dfrac{{81}}{{80}}} \right)\] .
A.0
B.1
C.\[\log 2\]
D.\[\log 3\]

Answer 1

Hint: Rewrite the given expression using the properties of logarithm. Substitute 16 by \[{2^4}\] ,15 by \[3 \times 5\], 25 by \[{5^2}\], 24 by \[{2^3} \times 3\], 81 by \[{3^4}\] and 80 by \[{2^4} \times 5\] in the obtained expression and calculate to obtain the required answer.

Formula Used: Some properties of logarithm are,
1.\[\log {a^b} = b\log a\]
2.\[\log a + \log b = \log ab\]

Complete step by step solution: The given expression is,
\[7\log \left( {\dfrac{{16}}{{15}}} \right) + 5\log \left( {\dfrac{{25}}{{24}}} \right) + 3\log \left( {\dfrac{{81}}{{80}}} \right)\]
=\[\log {\left( {\dfrac{{16}}{{15}}} \right)^7} + \log {\left( {\dfrac{{25}}{{24}}} \right)^5} + \log {\left( {\dfrac{{81}}{{80}}} \right)^3}\]
=\[\log \left[ {{{\left( {\dfrac{{16}}{{15}}} \right)}^7} \times {{\left( {\dfrac{{25}}{{24}}} \right)}^5} \times {{\left( {\dfrac{{81}}{{80}}} \right)}^3}} \right]\]
= \[\log \left[ {{{\left( {\dfrac{{{2^4}}}{{3 \times 5}}} \right)}^7} \times {{\left( {\dfrac{{{5^2}}}{{{2^3} \times 3}}} \right)}^5} \times {{\left( {\dfrac{{{3^4}}}{{{2^4} \times 5}}} \right)}^3}} \right]\]
\[ = \log \left[ {\dfrac{{{2^{28}} \times {5^{10}} \times {3^9}}}{{{3^7} \times {5^7} \times {2^{15}} \times {3^5} \times {2^{12}} \times {5^3}}}} \right]\]
\[ = \log \left[ {\dfrac{{{2^{28}} \times {5^{10}} \times {3^{12}}}}{{{2^{27}} \times {5^{10}} \times {3^{12}}}}} \right]\]
\[ = \log 2\]

Option ‘C’ is correct

Additional Information: If the base of two logarithmic expressions is not the same then we are unable to apply the product or quotient to them.
If the base of the logarithm expression is e then the logarithm is known as natural logarithem.
If the base of the logarithm expression is 10 then the logarithm is known as a common loagarithm.

Note: Sometime students get confused while dealing with the indices of the numbers. Here, we are using the formula of indices \[{\left( {{x^m}} \right)^n} = {x^{mn}}\] to obtain the simplified values.