Ta có 7 mũ 714 > 2 mũ 1993

=> 2 mũ 1993 < 7 mũ 714

2¹² = 1025 ; 7³ = 343 \(\Rightarrow\) 2¹⁰ < 3.7³ \(\Rightarrow\)\(\left(2^{10}\right)^{238}< 3^{238}.\left(7^3\right)^{238}\)\(\Rightarrow\)2²³⁸⁰ < 3²³⁸.7⁷¹⁴ .

2⁸ = 256 ; 3⁴ = 243 => 3⁵ < 2^8

Ta có : 3³²⁸ = 3³.2²²⁵ = \(3^3.\left(3^5\right)^{47}< 3^3.\left(2^8\right)^{47}< 2^5.2^{376}\Rightarrow3^{328}< 2^{381}\)

2²³⁸⁰ < 2²³⁸.7⁷¹⁴ => 2²³⁸⁰ < 2²³⁸.7⁷¹⁴ => 2¹⁹⁹⁹ < 714 mà 2¹⁹⁹⁹ > 2¹⁹⁹³ => 2¹⁹⁹³ < 7⁷¹⁴ .

      \(2^{10}=1024< 1029=3.7^3\)

\(\Leftrightarrow\left(2^{10}\right)^{238}< \left(3.7^3\right)^{238}\)

\(\Leftrightarrow2^{2380}< 3^{238}.7^{714}\) \(\left(1\right)\)

      \(3^5=243< 256=2^8\) \(\left(2\right)\)

      \(3^3=27< 32=2^5\) \(\left(3\right)\)

      Từ \(\left(2\right)\), \(\left(3\right)\) ta có:

      \(3^{328}=3^3.3^{325}=3^3\left(3^5\right)^{47}< 2^5\left(2^8\right)^{47}=2^{381}\)\(\left(4\right)\)

      Từ \(\left(1\right)\), \(\left(4\right)\) ta có:

      \(2^{2380}< 3^{238}.7^{714}\)

\(\Leftrightarrow2^{2380}< 2^{381}.7^{714}\)

\(\Leftrightarrow2^{1999}< 7^{714}\)

\(\Leftrightarrow2^{1993}< 7^{714}\).

\(2^{10}=1024< 1025;7^3=343\Rightarrow2^{10}< 1025< 1029=3.7^3\)

\(\Rightarrow\left(2^{10}\right)^{238}< \left(3.7^3\right)^{238}\Leftrightarrow2^{2380}< 3^{238}.7^{714}\)

\(2^8=256;3^5=243\Rightarrow2^8>3^5\)

\(3^{328}=3^3.3^{325}=3^3.\left(3^5\right)^{47}< 3^3.\left(2^8\right)^{47}< 2^5.2^{376}=2^{381}\)\(\Rightarrow3^{328}< 2^{381}\)

\(2^{2380}< 3^{238}.7^{714}\Rightarrow2^{1999}< 7^{714}\Leftrightarrow2^{1993}< 7^{714}.\)