Question

1) 127127127127127​

Answer 1

Ответ:

$1)\ \ 5^{3\sqrt7-1}\cdot 5^{1-\sqrt7}:5^{2\sqrt7-1}=\dfrac{5^{(3\sqrt7-1)+(1-\sqrt7)}}{5^{2\sqrt7-1}}=\dfrac{5^{2\sqrt7}}{5^{2\sqrt7-1}}=\\\\\\=5^{(2\sqrt7)-(2\sqrt7-1)}=5^{2\sqrt7-2\sqrt7+1}=5^1=\boxed{\ 5\ }$

$2)\ \ \dfrac{5\, sin74^\circ }{cos37^\circ \cdot cos53^\circ }=\dfrac{5\cdot \overbrace{2\, sin37^\circ \cdot cos37^\circ }^{sin74^\circ }}{cos37^\circ \cdot cos53^\circ }=\dfrac{10\cdot sin37^\circ }{cos53^\circ } =\dfrac{10\cdot sin37^\circ }{cos(90^\circ -37^\circ )}=\\\\\\=\dfrac{10\cdot sin37^\circ }{sin37^\circ }=\boxed{\ 10\ }$

$3)\ \ log_5(x+3)+log_54=log_516\ \ ,\ \ \ ODZ:\ x+3>0\ ,\ \ x>-3\ ,\\\\\star \ \ log_{a}x+log_{b}y=log_{a}(xy)\ \ \star \\\\log_5(4(x+3))=log_516\\\\log_5(4x+12)=log_516\ \ \ \to \ \ \ 4x+12=16\ \ ,\ \ 4x=4\ \ ,\ \ \boxed{\ x=1\ }$

$4)\ \ log_2(5x-7)-log_25=log_221\ \ ,\ \ \ ODZ:\ 5x-7>0\ ,\ \ x>1,4\\\\\star \ \ log_{a}x-log_{a}y=log_{a}\frac{x}{y}\ \ \star \\\\log_2\dfrac{5x-7}{5}=log_221\ \ \ \to \ \ \ \dfrac{5x-7}{5}=21\ \ ,\ \ 5x-7=105\ \ ,\ \ 5x=112\ ,\\\\\boxed{\ x=22,4\ }$