Question

1. $2log_{}u + 4log_{}v$
2. $20log_{}a - 4log_{}b$
3. $log_{6} c + \frac{log_{6} a}{2} + \frac{log_{6} b}{2}$
4. $4log_{3} u + 3log_{3} v$
5. $5log_{7} x + 6log_{7} y + log_{7} z$
6. $2log_{4} 2 + 2log_{4} 11 - 4log_{4} 3$
7. $\frac{3ln 2}{2} + \frac{ln 3}{2} + \frac{ln 7}{2}$
8. $4log_{7} 11 + 16log_{7} 2 + 4log_{7} 3$
9. $log_{8} x + 4log_{8} y + 3log_{8} z$
10. $\frac{log_{3} a}{3} + \frac{log_{3} b}{3} + \frac{log_{3} c}{3} + \frac{log_{3} d}{3}$

Explanation:

11) Step1: Apply power rule

$2\log_b u = \log_b u^2$, $4\log_b v = \log_b v^4$

11) Step2: Apply product rule

$\log_b u^2 + \log_b v^4 = \log_b (u^2v^4)$

12) Step1: Apply power rule

$20\log_4 a = \log_4 a^{20}$, $4\log_4 b = \log_4 b^4$

12) Step2: Apply quotient rule

$\log_4 a^{20} - \log_4 b^4 = \log_4 \frac{a^{20}}{b^4}$

13) Step1: Apply power rule

$\frac{\log_6 a}{2} = \log_6 a^{1/2}$, $\frac{\log_6 b}{2} = \log_6 b^{1/2}$

13) Step2: Apply product rule

$\log_6 c + \log_6 a^{1/2} + \log_6 b^{1/2} = \log_6 (c \cdot a^{1/2}b^{1/2}) = \log_6 c\sqrt{ab}$

14) Step1: Apply power rule

$4\log_3 u = \log_3 u^4$, $3\log_3 v = \log_3 v^3$

14) Step2: Apply product rule

$\log_3 u^4 + \log_3 v^3 = \log_3 (u^4v^3)$

15) Step1: Apply power rule

$5\log_7 x = \log_7 x^5$, $6\log_7 y = \log_7 y^6$

15) Step2: Apply product rule

$\log_7 x^5 + \log_7 y^6 + \log_7 z = \log_7 (x^5y^6z)$

16) Step1: Apply power rule

$2\log_4 2 = \log_4 2^2$, $2\log_4 11 = \log_4 11^2$, $4\log_4 3 = \log_4 3^4$

16) Step2: Apply product/quotient rule

$\log_4 2^2 + \log_4 11^2 - \log_4 3^4 = \log_4 \frac{2^2 \cdot 11^2}{3^4}$

17) Step1: Apply power rule

$\frac{3\ln 2}{2} = \ln 2^{3/2}$, $\frac{\ln 3}{2} = \ln 3^{1/2}$, $\frac{\ln 7}{2} = \ln 7^{1/2}$

17) Step2: Apply product rule

$\ln 2^{3/2} + \ln 3^{1/2} + \ln 7^{1/2} = \ln (2^{3/2} \cdot 3^{1/2} \cdot 7^{1/2}) = \ln \sqrt{2^3 \cdot 3 \cdot 7}$

18) Step1: Apply power rule

$4\log_7 11 = \log_7 11^4$, $16\log_7 2 = \log_7 2^{16}$, $4\log_7 3 = \log_7 3^4$

18) Step2: Apply product rule

$\log_7 11^4 + \log_7 2^{16} + \log_7 3^4 = \log_7 (11^4 \cdot 2^{16} \cdot 3^4)$

19) Step1: Apply power rule

$4\log_8 y = \log_8 y^4$, $3\log_8 z = \log_8 z^3$

19) Step2: Apply product rule

$\log_8 x + \log_8 y^4 + \log_8 z^3 = \log_8 (xy^4z^3)$

20) Step1: Apply power rule

$\frac{\log_3 a}{3} = \log_3 a^{1/3}$, $\frac{\log_3 b}{3} = \log_3 b^{1/3}$, $\frac{\log_3 c}{3} = \log_3 c^{1/3}$, $\frac{\log_3 d}{3} = \log_3 d^{1/3}$

20) Step2: Apply product rule

$\log_3 a^{1/3} + \log_3 b^{1/3} + \log_3 c^{1/3} + \log_3 d^{1/3} = \log_3 (a^{1/3}b^{1/3}c^{1/3}d^{1/3}) = \log_3 \sqrt[3]{abcd}$

Note: For problem 11, the base was not visible, so a general base $b$ was assumed.

Answer:

1. $\log_b u^2v^4$ (assuming base $b$ for the original logs)
2. $\log_4 \frac{a^{20}}{b^4}$
3. $\log_6 c\sqrt{ab}$
4. $\log_3 u^4v^3$
5. $\log_7 x^5y^6z$
6. $\log_4 \frac{2^2 \cdot 11^2}{3^4}$
7. $\ln \sqrt{2^3 \cdot 3 \cdot 7}$
8. $\log_7 11^4 \cdot 2^{16} \cdot 3^4$
9. $\log_8 xy^4z^3$
10. $\log_3 \sqrt[3]{abcd}$