Question

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Answer 1

Dados log 2 = 0,301, log 3 = 0,477, log 7 = 0,845 e log 11 = 1,041

a) 49ˣ = 231 => (7²)ˣ = 231 => 7²ˣ = 231 => log 7²ˣ = log 231 => log 7²ˣ = log 3.7.11 => log 7²ˣ = log 3 + log 7 + log 11 => 2x.log 7 = log 3 + log 7 + log 11 => 2x.0,845 = 0,477 + 0,845 + 1,041 => 1,69x = 2,363 => x = 2,363/1,69 => x = 1,4

b) 4ˣ = 2420 => 2²ˣ = 2².5.112 => log 2²ˣ = log 2².5.11² => 2x.log 2 = 2log 2 + log 5 + 2 log 11 => 2x.0,301 = 2.0,301 + log 5 + 2.1,041 => 0,602x = 0,602 + log 5 + 2,082 => 0,602x = 2,684 + log 5 , temos que 5 = 10/2, então log 5 = log 10/2 = log 10 - lo2 = 1 - 0,301 = 0,699. Logo 0,602x = 2,684 + 0,699 => 0,602x = 3.383 => x = 3,383/0,602=> x = 5,6

c) 25ˣ = 2156 => 5²ˣ =  2².7².11 => log 5²ˣ = log 2².7².11 => 2xlog 5 = log 2² + log 7² + log 11 => 2xlog 5 = 2log 2 + 2log 7 + log 11 => 2x.0,699 = 2.0,301 + 2.0,845 + 1,041 => 1,398x = 0,602 + 1,69 + 1,041 => 1,398x = 3,333 => x = 3,333/1,398 => x = 2,38

d) 121ˣ = 5000 => 11²ˣ = 2².5⁴ => log 11²ˣ = log 2³.5⁴ => 2xlog 11 = log 2³ + log 5⁴ => 2xlog 11 = 3log 2 + 4log 5 => 2x.1,041 = 3.0,301 + 4.0,699 => 2,082x = 0,903 + 2,796 => 2,082x = 3,699 => x 3,699/2,082 => x = 1,78