\[ \log_a(x\cdot y) = \log_a(x)+\log_a(y) \] \[ \log_a\left(\frac{x}{y}\right) = \log_a(x)-\log_a(y) \] \[ \log_a\left(x^b\right) = b\cdot \log_a(x) \]
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➀ \( \log_a\left(\frac{1}{xy}\right) \) Lösung
\( \quad \log_a\left(\frac{1}{xy}\right) = \log_a(1)-\log_a(xy) = \\
\quad\quad = 0-(\log_a(x)+\log_a(y)) = -\log_a(x)-\log_a(y) \)
\quad\quad = 0-(\log_a(x)+\log_a(y)) = -\log_a(x)-\log_a(y) \)
➁ \( \log_a\left(\sqrt{\frac{x}{y}}\right) \) Lösung
\( \quad \log_a\left(\sqrt{\frac{x}{y}}\right) = \log_a\left(\left(\frac{x}{y}\right)^{\frac{1}{2}}\right) \\
\quad\quad = \frac{1}{2}\cdot\log_a\left(\frac{x}{y}\right) = \frac{1}{2}(\log_a(x)-\log_a(y)) \)
\quad\quad = \frac{1}{2}\cdot\log_a\left(\frac{x}{y}\right) = \frac{1}{2}(\log_a(x)-\log_a(y)) \)
③ \( \log_a\left(\sqrt{x\sqrt{y}}\right) \) Lösung
\( \quad \log_a\left(\sqrt{x\sqrt{y}}\right) = \log_a\left((x\sqrt{y})^\frac{1}{2}\right) = \frac{1}{2}\cdot\log_a(x\sqrt{y}) \\
\quad\quad = \frac{1}{2}\left(\log_a(x)+\log_a(y^\frac{1}{2})\right) = \frac{1}{2}\left(\log_a(x)+\frac{1}{2}\log_a(y)\right) \)
\quad\quad = \frac{1}{2}\left(\log_a(x)+\log_a(y^\frac{1}{2})\right) = \frac{1}{2}\left(\log_a(x)+\frac{1}{2}\log_a(y)\right) \)