1. **Evaluate the following logarithms:** 1) $\log_2 32$ - Since $32 = 2^5$, $\log_2 32 = 5$ 2) $\log_{10} 1000$ - Since $1000 = 10^3$, $\log_{10} 1000 = 3$ 3) $\log_{81} 3$ - Note $81 = 3^4$, so $\log_{81} 3 = \frac{1}{4}$ because $81^{1/4} = 3$ 4) $\log_4 0.25$ - $0.25 = \frac{1}{4} = 4^{-1}$, so $\log_4 0.25 = -1$ 5) $\log_{10} \sqrt[3]{1000}$ - $\sqrt[3]{1000} = 10$, so $\log_{10} 10 = 1$ 6) $\log_3 243$ - $243 = 3^5$, so $\log_3 243 = 5$ 7) $\log_{343} 7$ - $343 = 7^3$, so $\log_{343} 7 = \frac{1}{3}$ 2. **Simplify the following expressions:** Recall logarithm rules: - $\log a + \log b = \log (ab)$ - $\log a - \log b = \log \left(\frac{a}{b}\right)$ - $k \log a = \log a^k$ 1) $\log 5 + \log 3 - \log 2 = \log \left(\frac{5 \times 3}{2}\right) = \log \frac{15}{2}$ 2) $\log \left(\frac{9}{14}\right) - \log \left(\frac{15}{16}\right) + \log \left(\frac{35}{24}\right)$ = $\log \left(\frac{9}{14} \times \frac{16}{15} \times \frac{35}{24}\right)$ = $\log \left(\frac{9 \times 16 \times 35}{14 \times 15 \times 24}\right)$ Simplify numerator and denominator: - Numerator: $9 \times 16 = 144$, $144 \times 35 = 5040$ - Denominator: $14 \times 15 = 210$, $210 \times 24 = 5040$ So expression = $\log 1 = 0$ 3) $2 \log \left(\frac{16}{15}\right) + \log \left(\frac{25}{24}\right) - \log \left(\frac{32}{27}\right)$ = $\log \left(\frac{16}{15}\right)^2 + \log \left(\frac{25}{24}\right) - \log \left(\frac{32}{27}\right)$ = $\log \left(\frac{256}{225} \times \frac{25}{24} \times \frac{27}{32}\right)$ Calculate numerator and denominator: - Numerator: $256 \times 25 \times 27 = 256 \times 675 = 172800$ - Denominator: $225 \times 24 \times 32 = 225 \times 768 = 172800$ So expression = $\log 1 = 0$ 4) $\frac{\log_4 64}{\log_9 81}$ - $64 = 4^3$, so $\log_4 64 = 3$ - $81 = 9^2$, so $\log_9 81 = 2$ - Expression = $\frac{3}{2}$ 5) $\log \left(\frac{225}{32}\right) - \log \left(\frac{25}{81}\right) + \log \left(\frac{64}{729}\right)$ = $\log \left(\frac{225}{32} \times \frac{81}{25} \times \frac{64}{729}\right)$ Simplify numerator and denominator: - Numerator: $225 \times 81 \times 64$ - Denominator: $32 \times 25 \times 729$ Calculate: - $225 = 15^2$, $81 = 9^2$, $64 = 8^2$ - $32 = 2^5$, $25 = 5^2$, $729 = 27^2$ Calculate product: - Numerator: $225 \times 81 = 18225$, $18225 \times 64 = 1,166,400$ - Denominator: $32 \times 25 = 800$, $800 \times 729 = 583,200$ Ratio = $\frac{1,166,400}{583,200} = 2$ So expression = $\log 2$ 6) $\frac{\log_7 25}{\log_7 5} = \frac{\log_5 8}{\log_5 2}$ - Left side: $\frac{\log_7 5^2}{\log_7 5} = \frac{2 \log_7 5}{\log_7 5} = 2$ - Right side: $\frac{\log_5 2^3}{\log_5 2} = \frac{3 \log_5 2}{\log_5 2} = 3$ - So equality is false unless reinterpreted; but as given, left = 2, right = 3 7) $\log \left(\frac{450}{32}\right) + \log \left(\frac{25}{128}\right) + \log \left(\frac{64}{25}\right) + \log \left(\frac{32}{25}\right)$ = $\log \left(\frac{450}{32} \times \frac{25}{128} \times \frac{64}{25} \times \frac{32}{25}\right)$ Simplify: - Cancel $25$ in numerator and denominator - Cancel $32$ and $64$ appropriately Calculate numerator and denominator: - Numerator: $450 \times 25 \times 64 \times 32$ - Denominator: $32 \times 128 \times 25 \times 25$ After cancellations, expression simplifies to $\log \left(\frac{450 \times 64}{128 \times 25}\right)$ Calculate: - $450 \times 64 = 28800$ - $128 \times 25 = 3200$ Ratio = $\frac{28800}{3200} = 9$ So expression = $\log 9$ 8) $\log \left(\frac{145}{8}\right) - 3 \log \left(\frac{3}{2}\right) + \log \left(\frac{54}{29}\right)$ = $\log \left(\frac{145}{8} \times \frac{54}{29} \times \left(\frac{3}{2}\right)^{-3}\right)$ = $\log \left(\frac{145}{8} \times \frac{54}{29} \times \frac{8}{27}\right)$ Calculate numerator and denominator: - Numerator: $145 \times 54 \times 8 = 145 \times 432 = 62,640$ - Denominator: $8 \times 29 \times 27 = 8 \times 783 = 6,264$ Ratio = $\frac{62,640}{6,264} = 10$ So expression = $\log 10 = 1$ 3. **Find $x$ if:** 1) $\log_2 (x - 3) = 3$ - Rewrite as $x - 3 = 2^3 = 8$ - So $x = 8 + 3 = 11$ 2) $\log_3 (x + 4) = 4$ - $x + 4 = 3^4 = 81$ - $x = 81 - 4 = 77$ 3) $\log_3 (x + 5) = 4$ - $x + 5 = 3^4 = 81$ - $x = 81 - 5 = 76$ 4) $\log_4 (3x - 5) = 0$ - $3x - 5 = 4^0 = 1$ - $3x = 6$ - $x = 2$ 5) $\log_2 \left(\frac{1}{2}\right) = x$ - $x = \log_2 2^{-1} = -1$ 6) $\log_4 x = \frac{1}{2}$ - $x = 4^{1/2} = 2$ **Final answers:** 1) 5, 3, $\frac{1}{4}$, -1, 1, 5, $\frac{1}{3}$ 2) $\log \frac{15}{2}$, 0, 0, $\frac{3}{2}$, $\log 2$, false equality, $\log 9$, 1 3) 11, 77, 76, 2, -1, 2