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Bài tập 96 trang 132 SGK Toán 12 NC

Bài tập 96 trang 132 SGK Toán 12 NC

Giải các hệ phương trình:

\(\begin{array}{*{20}{l}}
{a)\left\{ {\begin{array}{*{20}{l}}
{{{\log }_2}(x - y) = 5 - {{\log }_2}(x + y)}\\
{\frac{{\log x - \log 4}}{{\log y - \log 3}} =  - 1}
\end{array}} \right.}\\
{b)\left\{ {\begin{array}{*{20}{l}}
{2{{\log }_2}x - {3^y} = 15}\\
{{3^y}.{{\log }_2}x = 2{{\log }_2}x + {3^{y + 1}}}
\end{array}} \right.}
\end{array}\)

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Hướng dẫn giải chi tiết

 
 

a) Điều kiện 

\(\left\{ \begin{array}{l}
x > 0;y > 0\\
x - y > 0;x + y > 0
\end{array} \right. \Leftrightarrow x > y > 0\)

\(\begin{array}{*{20}{l}}
\begin{array}{l}
\left\{ {\begin{array}{*{20}{l}}
{{{\log }_2}(x - y) = 5 - {{\log }_2}(x + y)}\\
{\frac{{\log x - \log 4}}{{\log y - \log 3}} =  - 1}
\end{array}} \right.\\
 \Leftrightarrow \left\{ {\begin{array}{*{20}{l}}
{{{\log }_2}(x - y) + {{\log }_2}(x + y) = 5}\\
{\log \frac{x}{4} =  - \log \frac{y}{4}}
\end{array}} \right.
\end{array}\\
\begin{array}{l}
 \Leftrightarrow \left\{ {\begin{array}{*{20}{l}}
{{{\log }_2}({x^2} - {y^2}) = 5}\\
{\log \frac{{xy}}{{12}} = 1}
\end{array}} \right.\\
 \Leftrightarrow \left\{ {\begin{array}{*{20}{l}}
{{x^2} - {y^2} = 32}\\
{xy = 12}
\end{array}} \right.
\end{array}
\end{array}\)

Giải hệ bằng phương pháp thế ta được x = 6, y = 2

Vậy S = {(6;2)}

b) Điều kiện: x > 0.

Ta có nghiệm phương trình: 

\(\left\{ \begin{array}{l}
2u - v = 15(1)\\
u.v = 2u + 3v(2)
\end{array} \right.\)

Từ (1) suy ra v = 2u – 15, thay vào (2) ta được:

\(\begin{array}{*{20}{l}}
\begin{array}{l}
u(2u - 15) = 2u + 3(2u - 15)\\
 \Leftrightarrow 2{u^2} - 23u + 45 = 0
\end{array}\\
{ \Leftrightarrow \left\{ {\begin{array}{*{20}{l}}
{u = 9 \Rightarrow v = 3}\\
{u = \frac{5}{2} \Rightarrow v =  - 10(L)}
\end{array}} \right.}
\end{array}\)

\(\begin{array}{l}
\left\{ {\begin{array}{*{20}{l}}
{u = 9}\\
{v = 3}
\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{l}}
{\log _2^x = 9}\\
{{3^y} = 3}
\end{array}} \right.\\
 \Leftrightarrow \left\{ {\begin{array}{*{20}{l}}
{x = {2^5} = 512}\\
{y = 1}
\end{array}} \right.
\end{array}\)

Vậy S={(512;1)}

-- Mod Toán 12 HỌC247

 
Nếu bạn thấy hướng dẫn giải Bài tập 96 trang 132 SGK Toán 12 NC HAY thì click chia sẻ 
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