Question

[Nguyễn Thùy Linh]

Tìm n \(\in\)\(ℤ\)biết:

a) \(9.27^n=3^5\)

b) \(\left(2^3:4\right).2^n=4\)

c)\(3^2.3^4.3^n=3^7\)

d)\(2^{-1}.2^n+4.2^{n=9.2^5}\)

e)\(243\ge3^n\ge9.3^2\)

f)\(2^{n+3}.2^n=128\)

Answer 1

[Trần_Hiền_Mai]

a) \(9.27^n=3^5\Rightarrow3^2.\left(3^3\right)^n=3^5\)

\(\Rightarrow3^2.3^{3n}=3^5\Rightarrow3^{5n}=3^5\)

\(\Rightarrow5n=5\Rightarrow n=1\)

b)\(\left(2^3:4\right).2^n=4\Rightarrow\left(2^3:2^2\right).2^n=2^2\)

\(\Rightarrow2.2^n=2^2\Rightarrow2^{1+n}=2^2\)

\(\Rightarrow1+n=2\Rightarrow n=1\)

c)\(3^2.3^4.3^n=3^7\Rightarrow3^{6+n}=3^7\)

\(\Rightarrow6+n=7\Rightarrow n=1\)

d)\(2^{-1}.2^n+4.2^n=9.2^5\)

\(\Rightarrow2^n\left(2^{-1}+4\right)=3^2.2^5\)

\(\Rightarrow\)\(2^n\left(\frac{1}{2}+4\right)=3^2.2^5\)

\(\Rightarrow\)\(2^n.\frac{3^2}{2}=3^2.2^5\)

\(\Rightarrow\)\(2^{n-1}.3^2=3^2.2^5\)

\(\Rightarrow n-1=5\Rightarrow n=6\)

e)\(243\ge3^n\ge9.3^2\)

\(\Rightarrow3^5\ge3^n\ge3^2.3^2\)

\(\Rightarrow3^5\ge3^n\ge3^4\)

\(\Rightarrow5\ge n\ge4\Rightarrow5;4\)

f)\(2^{n+3}.2^n=128\)

\(\Rightarrow2^{n+3+n}=2^7\)

\(\Rightarrow2^{2n+3}=2^7\)

\(\Rightarrow2n+3=7\Rightarrow2n=4\Rightarrow n=2\)

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