Question

name: _______________
properties of logarithms: matching
directions: match each condensed logarithm on the left to its equivalent expanded form on the right using the properties of logarithms.

1. $log \frac{x^{4}}{y}$ a. $5(\log6 - \log11)$
2. $log(5xy)$ b. $\log5 + \logx + \logy$
3. $log(3 * 2^{3})$ c. $\log5 + \log8$
4. $log(6 * 11)$ d. $6\logx + 6\logy$
5. $log \frac{2^{4}}{5}$ e. $4\logx - \logy$
6. $log(xyz^{2})$ f. $4\log2 - \log5$
7. $log(\frac{6}{11})^{5}$ g. $\logx + \logy + 2\logz$
8. $log(5 * 8)$ h. $\log7 -2\log12$
9. $log(2 * 11 * 7)$ i. $\logx + 6\logy$
10. $log \frac{x}{y^{5}}$ j. $20\logx + 5\logy$
11. $log(x^{20} * y^{5})$ k. $\logx - 5\logy$
12. $log \frac{3}{8}$ l. $4\log3 + 4\log8$
13. $log \frac{x^{4}}{y^{20}}$ m. $\log2 + \log11 + \log7$
14. $log (xy)^{6}$ n. $4\logx - 20\logy$
15. $log \frac{7}{12^{2}}$ o. $\log6 + \log11$
16. $log(xy^{6})$ p. $\log3 - \log8$
17. $log(3^{4} * 8^{4})$ q. $\log3 + 3\log2$

Explanation:

Step1: Apply quotient/power rules

$\log \frac{x^4}{y} = \log x^4 - \log y = 4\log x - \log y$

Step2: Apply product rule

$\log(5xy) = \log5 + \log x + \log y$

Step3: Apply product/power rules

$\log(3 * 2^3) = \log3 + \log2^3 = \log3 + 3\log2$

Step4: Apply product rule

$\log(6 * 11) = \log6 + \log11$

Step5: Apply quotient/power rules

$\log \frac{2^4}{5} = \log2^4 - \log5 = 4\log2 - \log5$

Step6: Apply product/power rules

$\log(xyz^2) = \log x + \log y + \log z^2 = \log x + \log y + 2\log z$

Step7: Apply power/quotient rules

$\log(\frac{6}{11})^5 = 5\log(\frac{6}{11}) = 5(\log6 - \log11)$

Step8: Apply product rule

$\log(5 * 8) = \log5 + \log8$

Step9: Apply product rule

$\log(2 * 11 * 7) = \log2 + \log11 + \log7$

Step10: Apply quotient/power rules

$\log \frac{x}{y^5} = \log x - \log y^5 = \log x - 5\log y$

Step11: Apply product/power rules

$\log(x^{20} * y^5) = \log x^{20} + \log y^5 = 20\log x + 5\log y$

Step12: Apply quotient rule

$\log \frac{3}{8} = \log3 - \log8$

Step13: Apply quotient/power rules

$\log \frac{x^4}{y^{20}} = \log x^4 - \log y^{20} = 4\log x - 20\log y$

Step14: Apply power/product rules

$\log (xy)^6 = 6\log(xy) = 6\log x + 6\log y$

Step15: Apply quotient/power rules

$\log \frac{7}{12^2} = \log7 - \log12^2 = \log7 - 2\log12$

Step16: Apply product/power rules

$\log(xy^6) = \log x + \log y^6 = \log x + 6\log y$

Step17: Apply product/power rules

$\log(3^4 * 8^4) = \log3^4 + \log8^4 = 4\log3 + 4\log8$

Answer:

1. E. $4\log x - \log y$
2. B. $\log5 + \log x + \log y$
3. Q. $\log3 + 3\log2$
4. O. $\log6 + \log11$
5. F. $4\log2 - \log5$
6. G. $\log x + \log y + 2\log z$
7. A. $5(\log6 - \log11)$
8. C. $\log5 + \log8$
9. M. $\log2 + \log11 + \log7$
10. K. $\log x - 5\log y$
11. J. $20\log x + 5\log y$
12. P. $\log3 - \log8$
13. N. $4\log x - 20\log y$
14. D. $6\log x + 6\log y$
15. H. $\log7 - 2\log12$
16. I. $\log x + 6\log y$
17. L. $4\log3 + 4\log8$