Notes
Practice Problems
Expand the following logarithms.
\(\textbf{1)}\) \( \displaystyle\log_{b}\left(\frac{x^2z}{y^5}\right) \)
The answer is \( 2\log_{b}(x)+\log_{b}(z)-5\log_{b}(y) \)
\(\,\,\,\,\,\,\log_{b}\left(\frac{x^2z}{y^5}\right) \)
\(\,\,\,\,\,\,\log_{b}(x^2) + \log_{b}(z) – \log_{b}(y^5) \)
\(\,\,\,\,\,\,2\log_{b}(x) + \log_{b}(z) – 5\log_{b}(y) \)
\(\textbf{2)}\) \( \displaystyle\log_{5}\left(\frac{\sqrt[3]{z}}{25y^5}\right) \)
The answer is \( \frac{1}{3}\log_{5}(z)-5\log_{5}(y)-2 \)
\(\,\,\,\,\,\,\log_{5}\left(\frac{\sqrt[3]{z}}{25y^5}\right) \)
\(\,\,\,\,\,\,\log_{5}(\sqrt[3]{z}) – \log_{5}(25) – \log_{5}(y)^5 \)
\(\,\,\,\,\,\,\log_{5}(z)^{1/3} – \log_{5}(25) – \log_{5}(y)^5 \)
\(\,\,\,\,\,\,\frac{1}{3}\log_{5}(z) – \log_{5}(25) – 5\log_{5}(y) \)
\(\,\,\,\,\,\,\frac{1}{3}\log_{5}(z) – 2 – 5\log_{5}(y) \)
\(\textbf{3)}\) \( \displaystyle\log_{2}\left(\frac{b\sqrt{y}}{32}\right) \)
The answer is \( \log_{2}(b)+\frac{1}{2}\log_{2}(y)-5 \)
\(\,\,\,\,\,\,\log_{2}\left(\frac{b\sqrt{y}}{32}\right) \)
\(\,\,\,\,\,\,\log_{2}(b) + \log_{2}(\sqrt{y}) – \log_{2}(32) \)
\(\,\,\,\,\,\,\log_{2}(b) + \log_{2}(y)^{1/2} – \log_{2}(32) \)
\(\,\,\,\,\,\,\log_{2}(b) + \frac{1}{2}\log_{2}(y) – \log_{2}(2^5) \)
\(\,\,\,\,\,\,\log_{2}(b) + \frac{1}{2}\log_{2}(y) – 5 \)
\(\textbf{4)}\) \( \log_{5}\left(\frac{4\sqrt{x+3}}{x}\right) \)
The answer is \( \displaystyle2\log_{5}(2)+\frac{1}{2}\log_{5}(x+3)-\log_{5}(x) \)
\(\,\,\,\,\,\,\log_{5}\left(\frac{4\sqrt{x+3}}{x}\right) \)
\(\,\,\,\,\,\,\log_{5}\left(4\right) + \log_{5}\left(x+3\right)^{1/2} – \log_{5}\left(x\right)\)
\(\,\,\,\,\,\,\log_{5}\left(2\right)^2 + \log_{5}(x+3)^{1/2} – \log_{5}\left(x\right)\)
\(\,\,\,\,\,\,2\log_{5}\left(2\right) + \frac{1}{2} \log_{5}(x+3) – \log_{5}\left(x\right)\)
\(\textbf{5)}\) \( \displaystyle\log_{2}\left(\frac{x^3\sqrt{y}}{4z}\right) \)
The answer is \( \displaystyle3\log_{2}(x)+\frac{1}{2}\log_{2}(y)-\log_{2}(4)-\log_{2}(z) \)
\(\,\,\,\,\,\,\displaystyle\log_{2}\left(\frac{x^3\sqrt{y}}{4z}\right) \)
\(\,\,\,\,\,\,\displaystyle\log_{2}(x^3) + \log_{2}(\sqrt{y}) – \log_{2}(4) – \log_{2}(z)\)
\(\,\,\,\,\,\,\displaystyle3\log_{2}(x) + \frac{1}{2}\log_{2}(y) – \log_{2}(4) – \log_{2}(z)\)
\(\textbf{6)}\) \( \displaystyle\log_{3}\left(\frac{9y}{x^2}\right) \)
The answer is \( \displaystyle2\log_{3}(3)+\log_{3}(y)-2\log_{3}(x) \)
\(\,\,\,\,\,\,\displaystyle\log_{3}\left(\frac{9y}{x^2}\right) \)
\(\,\,\,\,\,\,\displaystyle\log_{3}(9) + \log_{3}(y) – \log_{3}(x^2)\)
\(\,\,\,\,\,\,\displaystyle2\log_{3}(3) + \log_{3}(y) – 2\log_{3}(x)\)
\(\textbf{7)}\) \( \displaystyle\log_{10}\left(x^2y^3\right) \)
The answer is \( \displaystyle2\log_{10}(x)+3\log_{10}(y) \)
\(\,\,\,\,\,\,\displaystyle\log_{10}\left(x^2y^3\right) \)
\(\,\,\,\,\,\,\displaystyle\log_{10}(x^2) + \log_{10}(y^3)\)
\(\,\,\,\,\,\,\displaystyle2\log_{10}(x) + 3\log_{10}(y)\)
\(\textbf{8)}\) \( \displaystyle\log_{6}\left(\frac{\sqrt{y}^5}{y^2}\right) \)
The answer is \( \displaystyle\frac{5}{2}\log_{6}(y)-2\log_{6}(y) \)
\(\,\,\,\,\,\,\displaystyle\log_{6}\left(\frac{\sqrt{y}^5}{y^2}\right) \)
\(\,\,\,\,\,\,\displaystyle\log_{6}(\sqrt{y}^5) – \log_{6}(y^2)\)
\(\,\,\,\,\,\,\displaystyle\frac{5}{2}\log_{6}(x) – 2\log_{6}(y)\)
\(\textbf{9)}\) \( \displaystyle\log_{4}\left(16\sqrt{y}\right) \)
The answer is \( \displaystyle2+\frac{1}{2}\log_{4}(y) \)
\(\,\,\,\,\,\,\displaystyle\log_{4}\left(16\sqrt{y}\right) \)
\(\,\,\,\,\,\,\displaystyle\log_{4}(16) + \log_{4}(\sqrt{y})\)
\(\,\,\,\,\,\,\displaystyle2 + \frac{1}{2}\log_{4}(y)\)
\(\textbf{10)}\) \( \displaystyle\log_{8}\left(\frac{64}{x^3y}\right) \)
The answer is \( \displaystyle2-3\log_{8}(x)-\log_{8}(y) \)
\(\,\,\,\,\,\,\displaystyle\log_{8}\left(\frac{64}{x^3y}\right) \)
\(\,\,\,\,\,\,\displaystyle\log_{8}(64) – \log_{8}(x^3) – \log_{8}(y)\)
\(\,\,\,\,\,\,\displaystyle2 – 3\log_{8}(x) – \log_{8}(y)\)
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