a)
[tex]3^{x+3} = \frac{1}{81}[/tex]
[tex]3^{x+3} = 3^{-4}[/tex]
x + 3 = -4
x = -4 - 3
x = -7
b)
[tex]8^{3x+1} = 64[/tex]
[tex]8^{3x+1} = 8^2[/tex]
3x + 1 = 2
3x = 2 - 1
3x = 1
x = 1/3
c)
[tex]81^{x-2} = \sqrt[4]{27}[/tex]
[tex]81^{x-2} = 27^{\frac{1}{4}}[/tex]
[tex](3^4)^{x-2} = (3^3)^{\frac{1}{4} }[/tex]
[tex]3^{4{x-8}} = 3^\frac{3}{4}[/tex]
4x - 8 = 3/4
4x = 0,75 + 8
4x = 8,75
x = [tex]\frac{8,75}{4}[/tex]
x = 2,1875
d)
[tex]\sqrt{16^{x+1}} = \sqrt[4]{64}[/tex]
[tex]16^{\frac{x+1}{2}} = 64^{\frac{1}{4}}[/tex]
[tex](2^4)^{\frac{x+1}{2}} = (2^6)^{\frac{1}{4}}[/tex]
[tex]2^{\frac{4x+4}{2}} = 2^{\frac{3}{2}}[/tex]
[tex]\frac{4x+4}{2} = \frac{3}{2} *(2)[/tex]
4x + 4 = 3
4x = 3 - 4
4x = -1
x = [tex]-\frac{1}4}[/tex] ou -0,25
e)
[tex]\sqrt{5^x} * 25^{x+1} = (0,2)^{1-x}[/tex]
[tex]5^\frac{x}{2} * (5^2)^{x+1} = (5^{-1})^{1-x}[/tex]
[tex]5^\frac{x}{2} * 5^{2x+2} = 5^{-1+x}[/tex]
[tex]\frac{x}{2}[/tex] + 2x + 2 = -1 + x
[tex]\frac{x}{2}[/tex] + 2x -x = -1 -2
[tex]\frac{x}{2}[/tex] + x = -3 (2)
x + 2x = -6
3x = -6
x = [tex]\frac{-6}{3}[/tex]
x = -2
f)
[tex]\sqrt[5]{2^x} * \sqrt[3]{4^x} = \sqrt{8^{-x}}[/tex]
[tex]\sqrt[5]{2^x} * \sqrt[3]{(2^2)^x} = \sqrt{(2^3)^{-x}}[/tex]
[tex]2^{\frac{x}{5}} * \sqrt[3]{2^{2x}} = \sqrt{2^{-3x}}[/tex]
[tex]2^{\frac{x}{5}} * 2^{\frac{2x}{3}} = 2^{\frac{-3x}{2}}[/tex]
[tex]\frac{x}{5} + \frac{2x}{3} = \frac{-3x}{2}[/tex]
mmc(5, 3, 2) = 30
[tex]\frac{6x}{30} + \frac{20x}{30} = \frac{-45x}{30}[/tex]
6x + 20x = -45x
26x + 45x = 0
71x = 0
x = 0
g)
[tex](\frac{2}{5})^{x+3} = (\frac{125}{8})^{x-1} * (0,4)^{2x-3}[/tex]
[tex](\frac{2}{5})^{x+3} = (\frac{5}{2})^{3*{(x-1)}} * (\frac{2}{5} )^{2x-3}[/tex]
[tex](\frac{2}{5})^{x+3} = (\frac{2}{5})^{(-1)*3*{(x-1)}} * (\frac{2}{5} )^{2x-3}[/tex]
[tex](\frac{2}{5})^{x+3} = (\frac{2}{5})^{-3x+3} * (\frac{2}{5} )^{2x-3}[/tex]
x+3 = -3x + 3 + 2x - 3
x + 3x - 2x = 3 - 3 - 3
2x = -3
x = [tex]-\frac{3}{2}[/tex]
h)
[tex](\frac{1}{27})^{-x} * (3^{3x})^2 = (\frac{1}{3})^{x-1}[/tex]
[tex](\frac{1}{3^3})^{-x} * 3^{6x} = (3^{-1})^{x-1}[/tex]
[tex](\frac{1}{3})^{-3x} * 3^{6x} = 3^{-x+1}[/tex]
[tex](3^{-1})^{-3x} * 3^{6x} = 3^{-x+1}[/tex]
[tex]3^{3x} * 3^{6x} = 3^{-x+1}[/tex]
3x + 6x = -x + 1
9x + x = 1
10x = 1
x = [tex]\frac{1}{10}[/tex]
