(1) \(f(x) = 0 \text{ dan } g(x) = 0\), dicari nilai \(x\) yang memenuhi kedua persamaan
\(2^{x\:-\: 3}=5^{x^2 + 2x \:-\: 15}\)
Langkah 1
\(f(x) = 0 \text{ dan } g(x) = 0\), dicari nilai \(x\) yang memenuhi kedua persamaan
\(x\:-\: 3 = 0\)
\(x = 3\:\:\:\:\:\color{blue}\text{memenuhi}\)
\(x^2 + 2x\:-\: 15 = 0\)
\((x + 5)(x \:-\: 3) = 0\)
\(x + 5 = 0 \rightarrow x_1 = - 5\:\:\:\:\:\color{red}\text{tidak memenuhi}\)
\(x \:-\: 3 = 0 \rightarrow x_2 = 3\:\:\:\:\:\color{blue}\text{memenuhi}\)
Yang memehuhi kedua persamaan adalah \(3\)
Langkah 2
Kedua ruas ditarik logaritma
\(\log 2^{x\:-\: 3}=\log 5^{x^2 + 2x \:-\: 15}\)
\(\color{blue} ^a\log b^{m} = m\cdot ^a \log b\)
\((x \:-\: 3)\log 2 = (x^2 + 2x \:-\: 15) \log 5\)
\(\dfrac{x\:-\: 3}{x^2 + 2x\:-\: 15} = \dfrac{\log 5}{\log 2}\)
\(\dfrac{\cancel{x\:-\: 3}}{(x + 5)\cancel{(x\:-\: 3)}} = \dfrac{\log 5}{\log 2}\)
\(x +5 = \dfrac{\log 2}{\log 5}\:\:\:\:\:\color{blue} \dfrac{\log a}{\log b} = ^b\log a\)
\(x + 5 = ^5\log 2\)
\(x = -5 + ^5\log 2\)
\(\bbox[yellow, 5px]{\text{HP} = \lbrace 3, \: -5 + ^5\log 2\rbrace}\)
Tentukan himpunan penyelesaian persamaan \((0,2)^{5\:-\: x}=6^{x^2 \:-\: 7x + 10}\)
\((0,2)^{5\:-\: x}=6^{x^2 \:-\: 7x + 10}\)
\(\Big(\dfrac{2}{10}\Big)^{5\:-\: x}=6^{(x\:-\: 5)(x\:-\: 2)}\)
\(\Big(\dfrac{1}{5}\Big)^{5\:-\: x}=6^{(x\:-\: 5)(x\:-\: 2)}\)
\(5^{-1(5\:-\: x)}=6^{(x\:-\: 5)(x\:-\: 2)}\)
\(5^{x\:-\: 5}=6^{(x\:-\: 5)(x\:-\: 2)}\)
Langkah 1
\(f(x) = 0 \text{ dan } g(x) = 0\), dicari nilai \(x\) yang memenuhi kedua persamaan
\(x\:-\: 5 = 0\)
\(x = 5\:\:\:\:\:\color{blue}\text{memenuhi}\)
\((x \:-\: 5)(x\:-\: 2) = 0\)
\(x \:-\: 5 = 0 \rightarrow x_1 = 5\:\:\:\:\:\color{blue}\text{memenuhi}\)
\(x \:-\: 2 = 0 \rightarrow x_2 = 2\:\:\:\:\:\color{red}\text{tidak memenuhi}\)
Yang memehuhi kedua persamaan adalah \(5\)
Langkah 2
Kedua ruas ditarik logaritma
\(\log 5^{(x\:-\: 5)}=\log 6^{(x\:-\: 5)(x\:-\: 2)}\)
\(\color{blue} ^a\log b^{m} = m\cdot ^a \log b\)
\((x\:-\: 5)\log 5 =(x\:-\: 5)(x\:-\: 2) \log 6\)
\(\dfrac{\cancel{(x\:-\: 5)}}{\cancel {(x\:-\: 5)}(x\:-\: 2)} = \dfrac{\log 6}{\log 5}\)
\(\dfrac{1}{x\:-\: 2} = \dfrac{\log 6}{\log 5}\)
\(x\:-\: 2 = \dfrac{\log 5}{\log 6}\)
\(\color{blue} \dfrac{\log a}{\log b} = ^b\log a\)
\(x\:-\: 2 = ^6\log 5\)
\(x = 2 + ^6\log 5 \)
\(\bbox[yellow, 5px]{\text{HP} = \lbrace 5, \: 2+^6\log 5 \rbrace}\)
\(8^{x\:-\: 6}=3^{2x^2 \:-\: 13x + 6}\)
\(2^{3(x\:-\: 6)}=3^{2x^2 \:-\: 13x + 6}\)
Langkah 1
\(f(x) = 0 \text{ dan } g(x) = 0\), dicari nilai \(x\) yang memenuhi kedua persamaan
\(3(x\:-\: 6) = 0\)
\(3x \:-\: 18 = 0\)
\(3x = 18\)
\(x = \dfrac{18}{3} = 6 \:\:\:\:\:\color{blue}\text{memenuhi}\)
\(2x^2 \:-\: 13x + 6 = 0\)
\((2x \:-\: 1)(x\:-\: 6) = 0\)
\(2x\:-\: 1 = 0 \rightarrow x_1 = \frac{1}{2}\:\:\:\:\:\color{red}\text{tidak memenuhi}\)
\(x \:-\: 6 = 0 \rightarrow x_2 = 6\:\:\:\:\:\color{blue}\text{memenuhi}\)
Yang memehuhi kedua persamaan adalah \(6\)
Langkah 2
Kedua ruas ditarik logaritma
\(\log 8^{x\:-\: 6}=\log 3^{2x^2 \:-\: 13x + 6}\)
\((x\:-\: 6)\log 8 = (2x^2 \:-\: 13x + 6)\log 3\)
\(\dfrac{(x\:-\: 6)}{2x^2 \:-\: 13x + 6} = \dfrac{\log 3}{\log 8}\)
\(\dfrac{\cancel{(x\:-\: 6)}}{(2x\:-\: 1)\cancel{(x-6)}} = \dfrac{\log 3}{\log 8}\)
\(\dfrac{1}{(2x\:-\: 1)} = \dfrac{\log 3}{\log 8}\)
\(2x\:-\: 1 = \dfrac{\log 8}{\log 3}\)
\(2x = 1+ ^3\log 8\)
\(2x = ^3\log 3 + ^3\log 8\:\:\:\:\:\color{blue} ^a\log b + ^a \log c = ^a \log bc\)
\(2x = ^3\log 24\)
\(x = \frac{1}{2}\cdot ^3\log 24\:\:\:\:\:\color{blue} \frac{m}{n}\cdot ^a\log b = ^{a^n}\log b^m \)
\(x = ^{3^{2}}\log 24^1\)
\(x = ^9\log 24\)
\(\bbox[yellow, 5px]{\text{HP} = \lbrace 6, \:^9\log 24\rbrace}\)
