Example 2.51: 1. Express in terms of simpler logarithms. a) log_(b)(x^3y) b) log_(b)(x^3+y) c) log_(b)((sqrt (xy))/(z^3))

By using the properties of logarithms, we can simplify each problem:<br /><br />a) \(\log _{b}\left(x^{3} y\right) \)<br />We can turn this into a simpler logarithm by using the "product rule" of logarithms, i.e. \( \log_{b}(AB) = \log_{b}(A) + \log_{b}(B) \) and the "power rule" of logarithms, i.e., \( \log_{b}(A^{n}) = n*\log_{b}(A) \).<br />We get,<br /><br />\( \log_{b}(x^{3} y) = \log_{b}(x^{3}) + \log_{b}(y) = 3 \log_{b}(x) + \log_{b}(y)\)<br /><br />b) \( \log _{b}\left(x^{3}+y\right) \)<br />This case already took the simplest form. The rule of logarithm that could separate thesis logarithms for sums doesn't apply here since the Table added to non‑logarithm. So,<br /><br />\( \log _{b}\left(x^{3}+y\right) \) stays as is.<br /><br />c) \( \log _{b}\left(\frac{\sqrt{x y}}{z^{3}}\right) \)<br />As in part a), we will use both the product and power rules, as well as the "quotient rule", i.e., \( \log_{b}(A / B) = \log_{b}(A) - \log_{b}(B) \).<br />We get,<br /><br />\( \log _{b}\left(\frac{\sqrt{x y}}{z^{3}}\right) = \log_{b}(\sqrt{x y}) - \log_{b}(z^{3})<br />= \frac{1}{2}\left(\log_{b}(x) + \log_{b}(y)\right) - 3 \log_{b}(z) \).