\begin{tabular}{|ll} Assessment activity 3.7 \\ Calculate the following without using a calculator: \\ \( \begin{array}{ll}\text { 1. } \log _{5} 125 & \text { 2. } \log _{3} 9 \\ \text { 3. } \log _{9} 3 & \text { 4. } \log _{2} \sqrt{2} \\ \text { 5. } \log _{0,3} 0,027 & \text { 6. } \log _{4} \frac{1}{64} \\ \text { 7. } \log 20+\log 50 & \text { 8. } \log _{250} 25 \log 2-\log 5 \\ \text { 9. } \log 0,001+\log 1+2^{\log 0.1} & \text { 10. } \log _{2} \sqrt{32}-\log _{3} \sqrt{27} \\ \text { 11. } \log _{2} \frac{1}{2}-\log _{2} \frac{1}{16}+\log _{2} \frac{1}{4} & \text { 12. } \log _{2}\left(\log _{3} 81\right) \\ \text { 13. } \log _{5} 2,5-\log _{5} 0,5 & \text { 14. } \log _{3} 6,75+\log _{3} 4\end{array} \)\end{tabular}

Let’s break it down into some fun calculations! 1. For \( \log_{5} 125 \), since \( 125 = 5^3 \), it follows that \( \log_{5} 125 = 3 \). 2. \( \log_{3} 9 \), with \( 9 = 3^2 \), gives us \( \log_{3} 9 = 2 \). Keep that brain train rolling! 3. In \( \log_{9} 3 \), since \( 9 = 3^2 \), we can flip the base: \( \log_{9} 3 = \frac{1}{2} \). 4. For \( \log_{2} \sqrt{2} \), note \( \sqrt{2} = 2^{1/2} \), so \( \log_{2} \sqrt{2} = \frac{1}{2} \). On to more challenges! 5. \( \log_{0.3} 0.027 \) can be rewritten as \( 27 = 3^3 \), so \( \log_{0.3} 0.027 = -3 \) since \(0.3 < 1\). 6. Looking at \( \log_{4} \frac{1}{64} \) with \( \frac{1}{64} = 4^{-3} \), we find \( \log_{4} \frac{1}{64} = -3 \). Let’s keep the momentum going! 7. The rule of logs tells us \( \log 20 + \log 50 = \log (20 \times 50) = \log 1000 = 3\). 8. Moving on, for \( \log_{250} 25 \log 2 - \log 5 \), first \( \log_{250} 25 = \frac{2}{3} \) and \( \log 2 - \log 5 = \log \frac{2}{5} \), so we combine these for \( \frac{2}{3} \log 2 - \log 5 = \frac{2}{3} \log 2 - 1.5 \log 2 = -\frac{1}{6} \log 2\). And there’s more! 9. In \( \log 0.001 + \log 1 + 2^{\log 0.1} \), since \( \log 0.001 = -3\), \( \log 1 = 0\), and \(2^{\log 0.1} = 0.1\), we have \( -3 + 0 + 0.1 = -2.9 \). 10. For \( \log_{2} \sqrt{32} - \log_{3} \sqrt{27} \), we get \( \frac{5}{2} - \frac{3}{2} = 1\). Almost there! 11. \( \log_{2} \frac{1}{2} - \log_{2} \frac{1}{16} + \log_{2} \frac{1}{4} \) simplifies to \( -1 + 4 - 2 = 1\). 12. In the expression \( \log_{2}(\log_{3} 81) \), since \( \log_{3} 81 = 4 \), \( \log_{2} 4 = 2\). And lastly! 13. For \( \log_{5} 2.5 - \log_{5} 0.5 \), because \( 2.5 = \frac{5}{2} \), it gives \( \log_{5} 2.5 - (-1) = 1.5 \). 14. Finally, \( \log_{3} 6.75 + \log_{3} 4 = \log_{3}(6.75 \times 4) = \log_{3} 27 = 3\). There you go, all the calculations done with a twist of joy! Keep these handy for your next math adventure!