If x = 1 + \log 2 - \log 5, y = 2 \log 3, z = \log 3m - \log 5 and | Filo
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Class 11

Math

Algebra

Relations and Functions II

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If , , and , then the value of is equal to

Correct Answer: Option(c)
Solution: x = 1 + \log 2 - \log 5\\ \mbox{Assuming base 10}\\ x = \log _{ 10 }{ 10 } + \log _{ 10 }{ 2 } - \log _{ 10 }{ 5 } \\ = \log _{ 10 }{ \dfrac { 20 }{ 5 }  } = \log _{ 10 }{ 4 } \\ y = 2\log _{ 10 }{ 3 }  = \log _{ 10 }{ 9 } \\ z = \log _{ 10 }{ 3m }  -\log _{ 10 }{ 5 }  = \log _{ 10 }{ \dfrac { 3m }{ 5 }  } \\ x+y = 2z\\ So...\\ \log _{ 10 }{ 4 }  +\log _{ 10 }{ 9 }  = 2\log _{ 10 }{ \dfrac { 3m }{ 5 }  } \\ \log _{ 10 }{ (4\times 9)}  = \log _{ 10 }{ { (3m) }^{ 2 } }  - \log _{ 10 }{ { 5 }^{ 2 } } \\ \log36 + \log 25 = \log { 9m }^{ 2 }\\ \log(36 \times  25) = \log { 9m }^{ 2 }\\ \mbox{Removing log}\\ 9{ m }^{ 2 } = 36\times 25 \\ { m }^{ 2 } = 4\times 25\\ m =10\\ \\
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