Question

Determine o conjunto verdade das equações exponenciais.

a)$5^{2x+1} = \frac{11}{625}$

b) 49^{2x+3} = 343

Answer 1

$a)~~5^{2x+1}=\dfrac{11}{625}\\\\\\\text{Vamos aplicar log de base 5 em ambos os lados da igualdade}\\\\\\log_5(5)^{2x+1}=log_5\begin{pmatrix}\dfrac{11}{625}\end{pmatrix}\\\\\\\star\text{O log de uma pot\^encia \'e igual ao produto do expoente pelo log}\\\text{da base da pot\^encia;}\\\\\star\text{O log do quociente \'e igual a diferen\c{c}a entre o log do dividendo}\\\text{e o log do divisor;}$

$(2x+1)\cdot log_5(5)=log_5(11)-log_5(625)\\\\(2x+1)\cdot1=log_5(11)-log_5(5^4)\\\\2x+1=log_5(11)-4\cdot log_5(5)\\\\2x+1=log_5(11)-4\cdot1\\\\2x+1=log_5(11)-4\\\\2x=log_5(11)-5\\\\\fbox{ x= \dfrac{log_5(11)-5}{2} }~\Leftrightarrow~S=\begin{Bmatrix}\dfrac{log_5(11)-5}{2}\end{Bmatrix}$


$b)~~49^{2x+3}=343\\\\\text{Vamos aplicar as propriedades exponenciais a fim de encontrar}\\\text{uma igualdade entre pot\^encias de mesma base;}\\\\\star\text{Lembrando que}~~\fbox{ (a^m)^n~\Leftrightarrow~a^{m\cdot n} }\\\\\\(7^2)^{2x+3}=7^3\\\\7^{4x+6}=7^3\\\\\\\star\text{Como}~~\fbox{ a^b=a^c~\Leftrightarrow~b=c }~~~~\text{para}~~~~(0\ \textless \ a \neq 1)\\\\\\4x+6=3\\\\4x=-3\\\\\fbox{ x= -\dfrac{3}{4} }~\Leftrightarrow~S=\begin{Bmatrix}-\dfrac{3}{4}\end{Bmatrix}$