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2)
![log(x + 1) = log(x - 1) + 3 \\ log(x + 1) = log(x - 1) + log( {10}^{3} ) \\ log(x + 1) = log(1000(x - 1)) \\ x + 1 = 1000(x - 1) \\ x + 1 = 1000x - 1000 \\ 1 + 1000 = 1000x - x \\ 1001 = 999x \\ x = \frac{1001}{999} log(x + 1) = log(x - 1) + 3 \\ log(x + 1) = log(x - 1) + log( {10}^{3} ) \\ log(x + 1) = log(1000(x - 1)) \\ x + 1 = 1000(x - 1) \\ x + 1 = 1000x - 1000 \\ 1 + 1000 = 1000x - x \\ 1001 = 999x \\ x = \frac{1001}{999}](https://tex.z-dn.net/?f=+log%28x+%2B+1%29++%3D++log%28x+-+1%29++%2B+3+%5C%5C++log%28x+%2B+1%29++%3D++log%28x+-+1%29++%2B++log%28+%7B10%7D%5E%7B3%7D+%29++%5C%5C++log%28x+%2B+1%29++%3D++log%281000%28x+-+1%29%29++%5C%5C+x+%2B+1+%3D+1000%28x+-+1%29+%5C%5C+x+%2B+1+%3D+1000x+-+1000+%5C%5C+1+%2B+1000+%3D+1000x+-+x+%5C%5C+1001+%3D+999x+%5C%5C+x+%3D++%5Cfrac%7B1001%7D%7B999%7D+)
3)
![2 log(x) - log(x + 6) = 0 \\ log( \frac{ {x}^{2} }{x + 6} ) = log(1) \\ \frac{ {x}^{2} }{x + 6} = 1 \\ {x}^{2} = x + 6 \\ {x}^{2} - x - 6 = 0 \\ (x - 3)(x + 2) 2 log(x) - log(x + 6) = 0 \\ log( \frac{ {x}^{2} }{x + 6} ) = log(1) \\ \frac{ {x}^{2} }{x + 6} = 1 \\ {x}^{2} = x + 6 \\ {x}^{2} - x - 6 = 0 \\ (x - 3)(x + 2)](https://tex.z-dn.net/?f=2+log%28x%29++-++log%28x+%2B+6%29++%3D+0+%5C%5C++log%28+%5Cfrac%7B+%7Bx%7D%5E%7B2%7D+%7D%7Bx+%2B+6%7D++%29++%3D++log%281%29++%5C%5C++%5Cfrac%7B+%7Bx%7D%5E%7B2%7D+%7D%7Bx+%2B+6%7D++%3D+1+%5C%5C++%7Bx%7D%5E%7B2%7D++%3D+x+%2B+6+%5C%5C++%7Bx%7D%5E%7B2%7D++-+x+-+6+%3D+0+%5C%5C+%28x+-+3%29%28x+%2B+2%29)
Sol 1: x=3
Sol2: x=-2
Pero como x>0
La solucion es x =3
4)
![log(x + 1) - log(x) = 1 \\ log( \frac{x + 1}{x} ) = log(10) \\ \frac{x + 1}{x} = 10 \\ x + 1 = 10x \\ x - 10x = - 1 \\ - 9x = - 1 \\ x = \frac{1}{9} log(x + 1) - log(x) = 1 \\ log( \frac{x + 1}{x} ) = log(10) \\ \frac{x + 1}{x} = 10 \\ x + 1 = 10x \\ x - 10x = - 1 \\ - 9x = - 1 \\ x = \frac{1}{9}](https://tex.z-dn.net/?f=+log%28x+%2B+1%29+++-++log%28x%29++%3D+1+%5C%5C++log%28+%5Cfrac%7Bx+%2B+1%7D%7Bx%7D+%29++%3D++log%2810%29++%5C%5C++%5Cfrac%7Bx+%2B+1%7D%7Bx%7D++%3D+10+%5C%5C+x+%2B+1+%3D+10x+%5C%5C+x+-+10x+%3D++-+1+%5C%5C++-+9x+%3D++-+1+%5C%5C+x+%3D++%5Cfrac%7B1%7D%7B9%7D+)
5)
![log(4x - 1) - log(x - 2) = log(5) \\ log( \frac{4x - 1}{x - 2} ) = log(5) \\ \frac{4x - 1}{x - 2} = 5 \\ 4x - 1 = 5(x - 2) \\ 4x - 1 = 5x - 10 \\ - 1 + 10 = 5x - 4x \\ 9 = x log(4x - 1) - log(x - 2) = log(5) \\ log( \frac{4x - 1}{x - 2} ) = log(5) \\ \frac{4x - 1}{x - 2} = 5 \\ 4x - 1 = 5(x - 2) \\ 4x - 1 = 5x - 10 \\ - 1 + 10 = 5x - 4x \\ 9 = x](https://tex.z-dn.net/?f=+log%284x+-+1%29++-++log%28x+-+2%29++%3D++log%285%29++%5C%5C++log%28++%5Cfrac%7B4x+-+1%7D%7Bx+-+2%7D++%29++%3D++log%285%29++%5C%5C++%5Cfrac%7B4x+-+1%7D%7Bx+-+2%7D++%3D+5+%5C%5C+4x+-+1+%3D+5%28x+-+2%29+%5C%5C+4x+-+1+%3D+5x+-+10+%5C%5C++-+1+%2B+10+%3D+5x+-+4x+%5C%5C+9+%3D+x)
6)
![log_{3}(x) = 4 \\ {3}^{4} = x \\ 81 = x log_{3}(x) = 4 \\ {3}^{4} = x \\ 81 = x](https://tex.z-dn.net/?f=+log_%7B3%7D%28x%29++%3D+4+%5C%5C++%7B3%7D%5E%7B4%7D++%3D+x+%5C%5C+81+%3D+x)
7)
![log_{5}(x) + log_{5}(30) = 3 \\ log_{5}(30x) = 3 \\ {5}^{3} = 30x \\ 125 = 30x \\ x = \frac{125}{30} \\ x = \frac{25}{6} log_{5}(x) + log_{5}(30) = 3 \\ log_{5}(30x) = 3 \\ {5}^{3} = 30x \\ 125 = 30x \\ x = \frac{125}{30} \\ x = \frac{25}{6}](https://tex.z-dn.net/?f=+log_%7B5%7D%28x%29++%2B++log_%7B5%7D%2830%29++%3D+3+%5C%5C++log_%7B5%7D%2830x%29++%3D+3+%5C%5C++%7B5%7D%5E%7B3%7D++%3D+30x+%5C%5C+125+%3D+30x+%5C%5C+x+%3D++%5Cfrac%7B125%7D%7B30%7D++%5C%5C+x+%3D+++%5Cfrac%7B25%7D%7B6%7D+)
3)
Sol 1: x=3
Sol2: x=-2
Pero como x>0
La solucion es x =3
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