Question

Determine a equação exponencial do 3 tipo

4(elevado a x) = 5.2(elevado a x) - 4

Answer 1

$4^x=5\cdot2^x-4\\ (2^2)^x=5\cdot2^x-4\\ (2^x)^2-5\cdot2^x+4=0\\\\ 2^x=y\\\\ y^2-5y+4=0\\\\ \Delta=b^2-4ac\\ \Delta=(-5)^2-4\cdot1\cdot4\\ \Delta=25-24\\ \Delta=1\\\\ y= \dfrac{-b\pm \sqrt{\Delta} }{2a}= \dfrac{-(-5)\pm \sqrt{1} }{2\cdot1}= \dfrac{5\pm1}{2}\begin{cases}y_1=2\\ y_2=3\end{cases}$

$2^x=y,~lembra???\\\\ 2^x=2~~~~~~~~~~~~~2^x=3~(nao~convem)\\ 2^x=2^1\\ \not2^x=\not2^1\\\\ x_1=1~~~~~~~~~~~~~~$

Portanto..

$\Large\boxed{\text{S}=\{1\}}$

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Por outro lado, se quiser usar logaritmos, podemos ter..

$2^x=3\\ \log(2^x)=\log(3)\\ \log(2)^x=\log(3)\\ x\cdot\log(2)=\log(3)\\\\ \log(2)=0,301~~e~~\log(3)=0,477\\\\ x\cdot0,301=0,477\\\\ x= \dfrac{0,477}{0,301}\\\\ x\approx1,58$

Aí se considerássemos usando logaritmos, a solução seria..

$\Large\boxed{\text{S}=\{1;~1,58\}}$