Question

preciso do cálculo

Exercícios:
1) Calcule:

a) log 5 x + log 25 x = 6

b) log 3 (2x – 1) = 4

c) log 3 (x 2 – 5x + 5) = 0

Answer 1

1)

a)

$log5x + log25x = 6 \\ \\ log5 + logx + log25 + logx = 6 \\ \\ log5 + logx + log {5}^{2} + logx = 6 \\ \\ log5 + logx +2 log5 + logx = 6 \\ \\ 3log5 + 2logx = 6 \\ \\ log {5}^{3} + log {x}^{2} = 6 \\ \\ log( {5}^{3} \sdot{x}^{2} ) = 6 \\ \\ log125 {x}^{2} = 6 \\ \\ {10}^{6} = 125 {x}^{2} \\ \\ {x}^{2} = \frac{ {10}^{6} }{125} \\ \\ x = \sqrt{ \frac{ {10}^{6} }{125} } \\ \\ x = \frac{ \sqrt{ {10}^{2} . {10}^{2}. {10}^{2} } }{ \sqrt{ {5}^{2} \times 5} } \\ \\ \\ \\ x = \frac{1000}{5 \sqrt{5} } \times \frac{ \sqrt{5} }{ \sqrt{5} } \\ \\ x = \frac{1000 \sqrt{5} }{5 \sqrt{25} } \\ \\ x = \frac{1000 \sqrt{5} }{5.5} \\ \\ x = \frac{1000 \sqrt{5} }{25} \\ \\ \large \boxed{ \green{x = 40 \sqrt{5} }}$

b)

$log3(2x - 1) = 4 \\ \\ log(6x - 3) = 4 \\ \\ {10}^{4} = 6x - 3 \\ \\ 10000 + 3 = 6x \\ \\ 6x = 10003\\ \\ \large \boxed{\green{x = \frac{10003}{6} }}$

ou

$log_{3}(2x-1)=4\\\\ 3^4=2x-1\\\\ 81=2x-1\\\\81+1=2x\\\\ 82=2x\\\\ x=\dfrac{82}{2}\\\\ \Large\boxed{\green{x=41}}$

c)

$log_{3}( {x}^{2} - 5x + 5) = 0 \\ \\ {3}^{0} = {x}^{2} - 5x + 5 \\ \\ 1 = {x}^{2} - 5x + 5 \\ \\ 0 = {x}^{2} - 5x + 5 - 1 \\ \\ 0 = {x}^{2} - 5x + 4 \\ \\ {x}^{2} - 5x + 4 = 0 \\ \\ \boxed{\Delta = {b}^{2} - 4ac} \\ \\ a = 1 \\ b = - 5 \\ c = 4 \\ \\ \Delta = {( - 5)}^{2} - 4(1)(4) \\ \\ \Delta = 25 - 16 \\ \\ \Delta = 9 \\ \\ Bhaskara \\ \\ \frac{ - b \pm \sqrt{ \: \Delta} }{2a} \\ \\ \frac{ - ( - 5) \pm \sqrt{9} }{2(1)} \\ \\ \frac{5 \pm3}{2} \\ \\ x1 = \frac{5 + 3}{2} = \frac{8}{2} = \large \boxed{ \green{4}} \\ \\ x2 = \frac{5 - 3}{2} = \frac{2}{2} = \large \boxed{ \green{1}}$

$\Large \boxed{\underline{\blue{\bf Bons\: Estudos!}}}\\\\ \Large \boxed{\underline{\bf26/05/2021}}$