Question:

The value of log0.0085+log3817\text {log}_{0.008}\sqrt{5}+\text{log}_{\sqrt{3}}81-7 is equal to

Updated On: Sep 26, 2024
  • 13\frac{1}{3}
  • 23\frac{2}{3}
  • 56\frac{5}{6}
  • 76\frac{7}{6}
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The Correct Option is C

Solution and Explanation

The given expression can be simplified as follows:

log0.0085+log3 ​​817log_{0.008}\sqrt{​5}​+log_{\sqrt{3}}\ ​​81−7
For log0.0085:log⁡_{0.008}\sqrt{5}​:

=log53(512)=log_{5^{−3}}​(5^{\frac{1}{2}})
=log53(532)=log_{⁡5^{−3}}(5^{−\frac{3}{2}})
=16=−\frac{1}{6}

For log3 81:log_{\sqrt{⁡3}}\ 81:
=log312(34)=log⁡_{3^{\frac{1}{2}}}(3^4)

=log312(32)=log⁡_{3^{\frac{1}{2}}}(3^2)
=2=2

Putting it all together:
(16)+87=16+1=56(−\frac{1}{6})+8−7=−\frac{1}{6}+1=\frac{5}{6}

Therefore, the required result is 56.\frac{5}{6}.

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