1. Sederhanakan bentuk berikut dan selesaikan: (a) $4^5 \times 4^3 \times 4^{-6} = 4^{5+3-6} = 4^{2} = 16$ (b) $5^4 \times 3^4 \times (-6)^4 = 625 \times 81 \times 1296 = 65610000$ (c) Sama seperti (a), hasilnya adalah 16 (d) $\frac{5^4 \times 3^4}{(-6)^4} = \frac{625 \times 81}{1296} = 39.0625$ 2. Ubah ke notasi akar: (a) $6^{2/3} = \sqrt[3]{6^2} = \sqrt[3]{36}$ (b) $\left(6^{2/3}\right)^2 = 6^{4/3} = \sqrt[3]{6^4} = \sqrt[3]{1296}$ (c) $\frac{3^{1/7} \times 3^{4/7}}{3^{3/7}} = 3^{(1/7+4/7-3/7)} = 3^{2/7} = \sqrt[7]{3^2} = \sqrt[7]{9}$ (d) $7^{2/5} + 9^{3/5}$ tetap dalam bentuk pangkat, tidak bisa disederhanakan lebih jauh dalam notasi akar. 3. Sederhanakan dan selesaikan pengakaran: (a) $10\sqrt5 + 2\sqrt5 - \sqrt5 = (10 + 2 - 1)\sqrt5 = 11\sqrt5$ (b) $\sqrt[3]{27} \times 5 \times \sqrt[3]{125} = 3 \times 5 \times 5 = 75$ (c) $4\sqrt{\sqrt{169}} = 4\sqrt{13} = 4 \times \sqrt{13}$ (d) $\frac{\sqrt{16}}{\sqrt[2]{4}} = \frac{4}{2} = 2$ 4. Ubahlah ke notasi logaritma lalu selesaikan: (a) $5^4 = 625$, jadi $\log 625 = 4 \log 5$ (b) $\sqrt[3]{64} = 64^{1/3} = 4$, jadi $\log 4$ (c) $4^5 \times 4^3 : 4^{-6} = 4^{14} = 268435456$, jadi $\log 268435456 = 14 \log 4$ (d) $\frac{3^{9/2}}{\sqrt{243}} = 3^{9/2} \times 3^{-5/2} = 3^2 = 9$, jadi $\log 9$ 5. Tentukan nilai $k$: (a) $\log 49 = 2 \log 7 \approx 2 \times 0.8451 = 1.6902$ (b) $\log 17 \approx 1.2304$ (c) $\log 5^8 = 8 \log 5 \approx 8 \times 0.6990 = 5.592$ (d) $\log {6 \over 2} = \log 3 \approx 0.4771$ 6. Dengan $a=100$ dan $b=50$: (a) $\log(ab) = \log(100 \times 50) = \log 5000 \approx 3.6990$ (b) $\log{a \over b} = \log 2 = 0.3010$ (c) $\log(a^2 b) = \log(10000 \times 50) = \log 500000 = 5.6990$ (d) $\log{a^2 \over b} = \log{10000 \over 50} = \log 200 = 2.3010$ 7. Tentukan $x$: (a) $\log x = 0.3010 \Rightarrow x = 10^{0.3010} = 2$ (b) $\log x = 1.2304 \Rightarrow x = 10^{1.2304} \approx 17$ (c) $\log x^2 = 1.7482 \Rightarrow x^2 = 10^{1.7482} \Rightarrow x = \sqrt{56.2} \approx 7.5$ (d) $\log x^2 = 2.6021 \Rightarrow x^2 = 10^{2.6021} \Rightarrow x = \sqrt{400} = 20$ 8. Tentukan $x$: (a) $x^5 = 50000 \Rightarrow x = 50000^{1/5} = 10$ (b) $100^x = 50000 \Rightarrow 10^{2x} = 5 \times 10^4 \Rightarrow 2x = \log 50000 \Rightarrow x = \frac{\log 50000}{2} = 2.8495$ (c) $x^{37} = 2500 \times (7.5)^{37}$ logaritma tidak langsung diselesaikan tanpa log. (d) $72^{x-0.1621} = 36 \Rightarrow (x-0.1621)\log 72 = \log 36 \Rightarrow x = \frac{\log 36}{\log 72} + 0.1621 = 0.5575 + 0.1621 = 0.7196$ 9. Selesaikan: (a) $6 \log 36 = 6 \times 1.5563 = 9.3378$ (b) $8 \log 512 = 8 \times 2.7093 = 21.6744$ (c) $e \log 100 = e \times 2 = 5.4366$ (d) $\ln 512 = 6.2383$ 10. Selesaikan: (a) $\log 10 = 1$ (b) $\log e = 0.4343$ (c) $\ln 10 = 2.3026$ (d) $\ln e = 1$ 11. Akar persamaan kuadrat $x^2 - 10x + 21 = 0$: Gunakan rumus kuadrat $x = \frac{10 \pm \sqrt{(-10)^2 - 4 \times 1 \times 21}}{2} = \frac{10 \pm \sqrt{100 - 84}}{2} = \frac{10 \pm 4}{2}$ Akar: $x_1 = 7$, $x_2 = 3$ 12. Akar $x^2 + 3x - 18 = 0$: $x = \frac{-3 \pm \sqrt{3^2 - 4 \times 1 \times (-18)}}{2} = \frac{-3 \pm \sqrt{9 + 72}}{2} = \frac{-3 \pm 9}{2}$ Akar: $x_1 = 3$, $x_2 = -6$ 13. $2x^2 - 12x = 24 \Rightarrow 2x^2 - 12x - 24 = 0 \Rightarrow x^2 - 6x - 12 = 0$ $x = \frac{6 \pm \sqrt{36 + 48}}{2} = \frac{6 \pm \sqrt{84}}{2} = 3 \pm \sqrt{21}$ 14. Persamaan kuadrat akar 3 dan -15: Rumus: $x^2 - (3 + (-15))x + (3)(-15) = 0 \Rightarrow x^2 + 12x - 45 = 0$ 15. Selesaikan $x^2 + 8 = 9x \Rightarrow x^2 - 9x + 8 = 0$ $x = \frac{9 \pm \sqrt{81 - 32}}{2} = \frac{9 \pm \sqrt{49}}{2} = \frac{9 \pm 7}{2}$ Akar: $x_1 = 8$, $x_2 = 1$