Bentuk \(a^{f(x)} = b^{f(x)}\)
Solusi
\(f(x) = 0\)
Dengan syarat a dan b tidak sama dengan 0
Contoh
\(5^{x^2 - x - 12} = 7^{x^2 - x - 12}\)
Satu-satunya solusi adalah \(x^2 - x - 12 = 0\)
\begin{equation*}
\begin{split}
& x^2 - x - 12 = 0 \\\\
& (x + 3) (x - 4) = 0 \\\\
& \bbox[5px, border: 2px solid magenta] {x = 4 \text{ atau } x = -3}
\end{split}
\end{equation*}
Bentuk \(a^{f(x)} = b \: ^{g(x)}\)
Solusi
Solusi
Menambahkan \(\log\) pada kedua ruas.
\( \log a^{f(x)} = \log b \: ^{g(x)}\)
\( f(x) \:.\: \log a = g(x) \:.\: \log b \)
Contoh:
\(5^{x^2 + x - 2} = 3^{x + 2}\)
Solusi
\begin{equation*}
\begin{split}
& 5^{x^2 + x - 2} = 3^{x + 2} \\\\
& \log 5^{x^2 + x - 2} = \log 3^{x + 2} \\\\
& (x^2 + x - 2) \log 5 = (x + 2) \log 3 \\\\
& (x + 2)(x - 1) \log 5 - (x + 2) \log 3 = 0\\\\
& (x + 2)[(x - 1) \log 5 - \log 3] = 0
\end{split}
\end{equation*}
Faktor 1
\(x + 2 = 0 \)
\(\bbox[5px, border: 2px solid magenta] {x = -2} \)
Faktor 2
\((x - 1) \log 5 - \log 3 = 0 \)
\((x - 1) \log 5 = \log 3 \)
\(x - 1 = \dfrac{\log 3}{\log 5}\)
\({\color {blue} \dfrac{\log b}{\log a} = \log_a b}\)
\(x - 1 = \log_5 3 \)
\(x = 1 + \log_5 3 \)
\(x = \log_5 5 + \log_5 3 \)
\(\bbox[5px, border: 2px solid magenta] {x = \log_5 15}\)