Question

Cрочно! Решите показательные уравнения!

$2^{x+4} -3*5^{x} =5^{x+1} -4*2^{x}$


$5^{3x+1} -4*100^{x} +5*80^{x} -4^{3x+1}$

Answer 1

1.

$2^{x+4}-3\cdot 5^{x}=5^{x+1}-4\cdot 2^{x}\\\\2^{x+4}+4\cdot 2^{x}=5^{x+1}+3\cdot 5^{x}\\\\2^{x}\cdot (2^4+4)=5^{x}\cdot (5+3)$

$\\\\2^{x}\cdot 20=5^{x}\cdot 8$

Делим на   $5^{x}\cdot 20$

$\\\\(\frac{2}{5})^{x}=\frac{8}{20}\\\\(\frac{2}{5})^{x}=\frac{2}{5}\\\\x=1$

2.

$5^{3x+1}-4\cdot 100^{x}+5\cdot 80^{x}-4^{3x+1}=0\\\\5^{3x}\cdot 5-4\cdot (4\cdot 25)^{x}+5\cdot (5\cdot 16)^{x}-4^{3x}\cdot 4=0\\\\5^{3x}\cdot 5-4\cdot 4^{x}\cdot (5)^{2x}+5\cdot 5^{x}\cdot 4^{2x}-4^{3x}\cdot 4=0$

Делим на $4^{3x}$

$5(\frac{5}{4})^{3x}-4\cdot (\frac{5}{4})^{2x} +5\cdot(\frac{5}{4})^{x} -4=0$

Разложим на множители:

$(5\cdot (\frac{5}{4})^{3x}+5\cdot(\frac{5}{4})^{x}) -(4\cdot (\frac{5}{4})^{2x} +4)=0\\\\5\cdot (\frac{5}{4})^{x}\cdot((\frac{5}{4})^{2x}+1) -4\cdot ((\frac{5}{4})^{2x} +1)=0\\\\((\frac{5}{4})^{2x}+1) \cdot (5\cdot (\frac{5}{4})^{x}-4)=0$

$(\frac{5}{4})^{2x}+1>0 \\\\5\cdot (\frac{5}{4})^{x}-4=0\\\\(\frac{5}{4})^{x}=\frac{4}{5}\\\\(\frac{5}{4})^{x}=(\frac{5}{4})^{-1}\\\\x=-1$