Question

Preciso do desenvolvimento dessas contas, façam o máximo que conseguirem, pois vale bastante ponto.
Já tem o gabarito.

Answer 1

1) $(3^x)^{x-1}=729$

$3^{x^2-x}=3^6$

$x^2-x-6=0$

$x=3$ ou $x=-2$

$S=\{-2,3\}$


2) $\sqrt{3}=27^x$

$3^{\frac{1}{2}}=(3^3)^x$

$3^{\frac{1}{2}}=3^{3x}$

$3x=\dfrac{1}{2}$

$x=\dfrac{1}{6}$

$S=\{\frac{1}{6}\}$


3) $10^{2+x}=\sqrt[x]{1000^5}$

$10^{2+x}=\sqrt[xx]{(10^3)^5}$

$10^{2+x}=\sqrt[x]{10^{15}}$

$10^{2+x}=10^{\frac{15}{x}}$

$2+x=\dfrac{15}{x}$

$x^2+2x-15=0$

$x=3$ ou $x=-5$

$S=\{-5,3\}$


4) $\dfrac{1}{0,25}=(512)^{-x+1}$

$4=(2^9)^{-x+1}$

$2^2=2^{-9x+9}$

$-9x+9=2$

$-9x=-7$

$x=\dfrac{7}{9}$

$S=\{\frac{7}{9}\}$


5) $\dfrac{1}{\sqrt[5]{81}}=9^x$

$\dfrac{3^0}{\sqrt[5]{3^4}}=(3^2)^x$

$\dfrac{3^0}{3^{\frac{4}{5}}}=3^{2x}$

$3^{-\frac{4}{5}}=3^{2x}$

$2x=-\dfrac{4}{5}$

$x=-\dfrac{4}{10}=-\dfrac{2}{5}$

$S=\{\frac{-2}{5}\}$


6) $(0,7)^{3x+1}=\dfrac{343}{1000}$

$\left(\dfrac{7}{10}\right)^{3x+1}=\left(\dfrac{7}{10}\right)^3$

$3x+1=3$

$3x=2$

$x=\dfrac{2}{3}$

$S=\{\frac{2}{3}\}$


7) $\left[\left(\dfrac{5}{3}\right)^x\right]^x=\dfrac{625}{81}$

$\left(\dfrac{5}{3}\right)^{x^2}=\left(\dfrac{5}{3}\right)^4$

$x^2=4$

$x=\pm2$

$S=\{-2,2\}$


8) $0,125=\left(\dfrac{1}{\sqrt{2}}\right)^{x-1}$

$\dfrac{1}{8}=\left(\dfrac{1}{\sqrt{2}}\right)^{x-1}$

$\left(\dfrac{1}{\sqrt{2}}\right)^{6}=\left(\dfrac{1}{\sqrt{2}}\right)^{x-1}$

$x-1=6$

$x=7$

$S=\{7\}$


9) $2^{x}+2^{x-1}=48$

$2^{x}+2^{x}\cdot2^{-1}=48$

$2^{x}=y$:

$y+\dfrac{y}{2}=48$

$2y+y=96$

$3y=96$

$y=32$

$2^5=32$

$2^x=2^5$

$x=5$

$S=\{5\}$


10) $2^{x+4}-2^{x+1}=56$

$2^x\cdot2^{4}-2^{x}\cdot2=56$

$2^x=y$

$16y-2y=56$

$14y=56$

$y=4$

$2^x=4$

$2^x=2^2$

$x=2$

$S=\{2\}$.