Matematică, întrebare adresată de melissa1441, 8 ani în urmă

Ma ajuta cineva plz

Anexe:

Răspunsuri la întrebare

Răspuns de Seethh
1

\displaystyle I.~x^x=a^r \Leftrightarrow x=r \\\\1)~10^x=10^2 \Rightarrow x=2\\\\ 2)~4^x=\frac{1}{4} \Rightarrow 4^x=4^{-1}\Rightarrow x=-1\\\\ 3)~64^x=32 \Rightarrow 2^{6x}=2^5 \Rightarrow 6x=5 \Rightarrow x=\frac{5}{6} \\\\ 4)~3^x=\sqrt[4]{27} \Rightarrow 3^x=\sqrt[4]{3^3} \Rightarrow 3^x=3^{\frac{3}{4} }\Rightarrow x=\frac{3}{4} \\\\ 5)~6^x=1 \Rightarrow 6^x=1^0 \Rightarrow x=0

\displaystyle 6)~10^x=5^x \Rightarrow 2^x \cdot 5^x=5^x \Rightarrow 2^x=\frac{5^x}{5^x} \Rightarrow 2^x=1 \Rightarrow 2^x=1^0 \Rightarrow x=0\\\\ 7)~3^x=2^x \Rightarrow xln3=xln2 \Rightarrow xln3-xln2=0 \Rightarrow x(ln3-ln2)=0 \Rightarrow \\\\ \Rightarrow x=\frac{0}{ln3-ln2} \Rightarrow x=0

\displaystyle 8)~\sqrt{5^x} =\sqrt[3]{625} \Rightarrow \sqrt{5^x} =\sqrt[3]{5^4} \Rightarrow 5^{\frac{x}{2} }=5^{\frac{4}{3} }\Rightarrow \frac{x}{2} =\frac{4}{3} \Rightarrow 3x=4 \cdot 2 \Rightarrow \\\\ \Rightarrow 3x=8 \Rightarrow x=\frac{8}{3} \\\\ 9)~2^{2^x}=16 \Rightarrow 2^{2^x}=2^{2^2}\Rightarrow  2^x=2^2 \Rightarrow x=2

\displaystyle II.~a^{f(x)}=a^r \Leftrightarrow f(x)=r\\\\ 1)~16^x=64 \Rightarrow 2^{4x}=2^6 \Rightarrow 4x=6 \Rightarrow x=\frac{6}{4} \Rightarrow x=\frac{3}{2} \\\\ 2)~32^x=8 \Rightarrow 2^{5x}=2^3 \Rightarrow 5x=3 \Rightarrow x=\frac{3}{5} \\\\ 3)~8^x-2=0 \Rightarrow 8^x=0+2 \Rightarrow 2^{3x}=2 \Rightarrow 3x=1  \Rightarrow x=\frac{1}{3}

\displaystyle 4)~9^x-3=0 \Rightarrow 9^x=0+3 \Rightarrow 3^{2x}=3 \Rightarrow  2x=1 \Rightarrow x=\frac{1}{2} \\\\ 5)~16^{-x}-64=0 \Rightarrow 16^{-x}=0+64 \Rightarrow 2^{-4x}=2^6 \Rightarrow -4x=6 \Rightarrow x=-\frac{6}{4} \Rightarrow \\\\ \Rightarrow x=-\frac{3}{2}

\displaystyle 6)~27^{x+1}=3 \Rightarrow 3^{3(x+1)}=3 \Rightarrow 3^{3x+3}=3 \Rightarrow 3x+3=1 \Rightarrow 3x=1-3 \Rightarrow \\\\ \Rightarrow 3x=-2 \Rightarrow x=-\frac{2}{3}\\\\ 7)~3^{9x}-9^{2007}=0 \Rightarrow 3^{9x}=3^{2\cdot 2007}\Rightarrow 3^{9x}=3^{4014}\Rightarrow 9x=4014 \Rightarrow \\\\ \Rightarrow x=\frac{4014}{9} \Rightarrow x=446

\displaystyle 8)~2^{x^2+4}=4 \Rightarrow 2^{x^2+4}=2^2 \Rightarrow x^2+4=2 \Rightarrow x^2=2-4 \Rightarrow x^2=-2 \Rightarrow x\not\in\mathbb{R}\\\\ 9)~2007^{2x^2-1}=2007^7 \Rightarrow 2x^2-1=7 \Rightarrow 2x^2=7+1 \Rightarrow 2x^2=8 \Rightarrow \\\\ \Rightarrow x^2=\frac{8}{2} \Rightarrow x^2=4\Rightarrow x=\pm \sqrt{4} \Rightarrow x=\pm 2

\displaystyle III.~a^{f(x)}=a^{g(x)} \Leftrightarrow f(x)=g(x)\\\\ 1)~3^{-2x-1}=3^{x^2}\Rightarrow -2x-1=x^2 \Rightarrow -x^2-2x-1=0\big| \cdot(-1) \Rightarrow \\\\ \Rightarrow x^2+2x+1=0 \\\\ \Delta=2^2-4 \cdot 1 \cdot 1=4-4=0\\\\ x_1=x_2=-\frac{b}{2a} =-\frac{2}{2 \cdot 1} =-\frac{2}{2} =-1

\displaystyle 2)~4^x=8^x \Rightarrow 4^x= 2 ^x \cdot 4^x \Rightarrow 2^x=\frac{4^x}{4^x}\Rightarrow 2^x=1 \Rightarrow 2^x=1^0 \Rightarrow x=0\\\\ 3)~16^x=32^x \Rightarrow 16^x=2^x \cdot 16^x \Rightarrow 2^x=\frac{16^x}{16^x} \Rightarrow 2^x=1 \Rightarrow 2^x=1^0 \Rightarrow x=0

\displaystyle 7)~9^x=3^{x+1}\Rightarrow 3^{2x}=3^{x+1}\Rightarrow 2x=x+1 \Rightarrow 2x-x=1 \Rightarrow x=1\\\\ 8)~(0,25)^{3-x}=\frac{256}{2^{x+5}} \Rightarrow \Bigg(\frac{25}{100} \Bigg)^{3-x}=\frac{2^8}{2^{x+5}}  \Rightarrow \frac{1^{3-x}}{4^{3-x}} =\frac{2^8}{2^{x+5}} \Rightarrow \\\\ \Rightarrow 2^{8-x-5}=2^{2(3-x)}\Rightarrow 2^{-x+4}=2^{6-2x}\Rightarrow -x+3=6-2x \Rightarrow \\\\ \Rightarrow -x+2x=6-3 \Rightarrow x=3

\displaystyle IV. \\\\ 1)~3^{x+1}+3^x=108 \Rightarrow 3^x(3+1)=108 \Rightarrow 3^x \cdot 4=108 \Rightarrow 3^x=\frac{108}{4} \Rightarrow \\\\ \Rightarrow 3^x=27 \Rightarrow 3^x=3^3 \Rightarrow x=3\\\\ 2)~5^x+5^{x+1}=3750 \Rightarrow 5^x(1+5)=3750 \Rightarrow 5^x\cdot 6=3750 \Rightarrow 5^x=\frac{3750}{6} \Rightarrow \\\\ \Rightarrow 5^x=625 \Rightarrow 5^x=5^4 \Rightarrow x=4

\displaystyle 3)~2^{x+3}+2^{x+2}+2^{x+1}+2^x=30 \Rightarrow 2^x\Big(2^3+2^2+2^1+1\Big)=30 \Rightarrow \\\\ \Rightarrow 2^x(8+4+2+1)=30 \Rightarrow 2^x \cdot 15=30 \Rightarrow 2^x=\frac{30}{15} \Rightarrow 2^x=2 \Rightarrow x=1

\displaystyle 4)~7^x-7^{x-1}=6 \Rightarrow 7^x\Big(1-7^{-1}\Big)=6 \Rightarrow 7^x\Bigg(1-\frac{1}{7} \Bigg)=6 \Rightarrow \\\\ \Rightarrow 7^x \cdot \frac{7-1}{7} =6\Rightarrow 7^x\cdot \frac{6}{7} =6 \Rightarrow 7^x=\frac{6}{\cfrac{6}{7} }  \Rightarrow 7^x=6 \cdot \frac{7}{6} \Rightarrow 7^x=7 \Rightarrow x=1

\displaystyle 5)~7^{x+2}+4 \cdot 7^{x-1}=347 \Rightarrow 7^x\Big(7^2+4 \cdot 7^{-1}\Big)=347 \Rightarrow 7^x\Bigg(49+\frac{4}{7} \Bigg)=347 \Rightarrow \\\\ \Rightarrow 7^x \cdot \frac{343+4}{7}=347 \Rightarrow 7^x \cdot \frac{347}{7} =347 \Rightarrow 7^x=\frac{347}{\cfrac{347}{7} }  \Rightarrow 7^x=347 \cdot \frac{7}{347} \Rightarrow \\\\ \Rightarrow 7^x=7 \Rightarrow x=1


melissa1441: Mulțumesc frumos
Alte întrebări interesante