Question

determine o conjunto verdade das equações :
a) log de (x+20) na base x igual a 2
b) log De (2x+3) igual a 2

Answer 1

a)
$log_x(x+20)=2\\ \\ x^2=x+20\\ \\ x^2-x-20=0\\ \\ \Delta=(-1)^2-4.1.(-20)=1+80=81\\ \\ x=\frac{1\pm9}{2}\\ \\ x_1=-4\\ \\ x_2=5$
Logicamente que o resultado negativo não atende as exigências do enunciado.

Logo S={5}

b)
$log(2x+3)=2\\ \\ 2x+3=10^2\\ \\ 2x+3=100\\ \\ 2x=100-3\\ \\ 2x=97\\ \\ x=\frac{97}{2}$

Answer 2

a) $\text{log}_x~(x+20)=2$

Lembre-se que, $\text{log}_b~a=x~~\Rightarrow~~b^x=a$.

Assim, $\text{log}_x~(x+20)=2~~\Rightarrow~~x^2=x+20$.

$x^2-x-20=0$

$\Delta=(-1)^2-4\cdot1\cdot(-20)=1+80=81~~\Rightarrow~~x=\dfrac{-(-1)\pm\sqrt{81}}{2}$

$x=\dfrac{1\pm9}{2}~~\Rightarrow~~x'=\dfrac{1+9}{2}=5$ e $x"=\dfrac{1-9}{2}=-4$ (não serve).

$S=\{5\}$

b) $\text{log}~(2x+3)=2$.

Neste caso, $\text{log}_{10}~(2x+3)=2~~\Rightarrow~~2x+3=10^2$.

$2x+3=100~~\Rightarrow~~2x=97~~\Rightarrow~~x=\dfrac{97}{2}$