4.4.4E: Propiedades logarítmicas (Ejercicios)
- Page ID
- 116848
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)sección 4.4 ejercicio
Simplificar a un solo logaritmo, usando propiedades logaritmo.
1. \(\log _{3} \left(28\right)-\log _{3} \left(7\right)\)
2. \(\log _{3} \left(32\right)-\log _{3} \left(4\right)\)
3. \(-\log _{3} \left(\dfrac{1}{7} \right)\)
4. \(-\log _{4} \left(\dfrac{1}{5} \right)\)
5. \(\log _{3} \left(\dfrac{1}{10} \right)+\log _{3} \left(50\right)\)
6. \(\log _{4} \left(3\right)+\log _{4} (7)\)
7. \(\dfrac{1}{3} \log _{7} \left(8\right)\)
8. \(\dfrac{1}{2} \log _{5} \left(36\right)\)
9. \(\log \left(2x^{4} \right)+\log \left(3x^{5} \right)\)
10. \(\ln \left(4x^{2} \right)+\ln \left(3x^{3} \right)\)
11. \(\ln \left(6x^{9} \right)-\ln \left(3x^{2} \right)\)
12. \(\log \left(12x^{4} \right)-\log \left(4x\right)\)
13. \(2\log \left(x\right)+3\log \left(x+1\right)\)
14. \(3\log \left(x\right)+2\log \left(x^{2} \right)\)
15. \(\log \left(x\right)-\dfrac{1}{2} \log \left(y\right)+3\log \left(z\right)\)
16. \(2\log \left(x\right)+\dfrac{1}{3} \log \left(y\right)-\log \left(z\right)\)
Utilice las propiedades de logaritmo para expandir cada expresión.
17. \(\log \left(\dfrac{x^{15} y^{13} }{z^{19} } \right)\)
18. \(\log \left(\dfrac{a^{2} b^{3} }{c^{5} } \right)\)
19. \(\ln \left(\dfrac{a^{-2} }{b^{-4} c^{5} } \right)\)
20. \(\ln \left(\dfrac{a^{-2} b^{3} }{c^{-5} } \right)\)
21. \(\log \left(\sqrt{x^{3} y^{-4} } \right)\)
22. \(\log \left(\sqrt{x^{-3} y^{2} } \right)\)
23. \(\ln \left(y\sqrt{\dfrac{y}{1-y} } \right)\)
24. \(\ln \left(\dfrac{x}{\sqrt{1-x^{2} } } \right)\)
25. \(\log \left(x^{2} y^{3} \sqrt[{3}]{x^{2} y^{5} } \right)\)
26. \(\log \left(x^{3} y^{4} \sqrt[{7}]{x^{3} y^{9} } \right)\)
Resuelve cada ecuación para la variable.
27. \(4^{4x-7} =3^{9x-6}\)
28. \(2^{2x-5} =7^{3x-7}\)
29. \(17\left(1.14\right)^{x} =19\left(1.16\right)^{x}\)
30. \(20\left(1.07\right)^{x} =8\left(1.13\right)^{x}\)
31. \(5e^{0.12t} =10e^{0.08t}\)
32. \(3e^{0.09t} =e^{0.14t}\)
33. \(\log _{2} \left(7x+6\right)=3\)
34. \(\log _{3} (2x+4)=2\)
35. \(2\ln \left(3{\rm x}\right)+3=1\)
36. \(4\ln \left(5x\right)+5=2\)
37. \(\log \left(x^{3} \right)=2\)
38. \(\log \left(x^{5} \right)=3\)
39. \(\log \left(x\right)+\log \left(x+3\right)=3\)
40. \(\log \left(x+4\right)+\log \left(x\right)=9\)
41. \(\log \left(x+4\right)-\log \left(x+3\right)=1\)
42. \(\log \left(x+5\right)-\log \left(x+2\right)=2\)
43. \(\log _{6} \left(x^{2} \right)-\log _{6} (x+1)=1\)
44. \(\log _{3} (x^{2} )-\log _{3} (x+2)=5\)
45. \(\log \left(x+12\right)=\log \left(x\right)+\log \left(12\right)\)
46. \(\log \left(x+15\right)=\log \left(x\right)+\log \left(15\right)\)
47. \(\ln \left(x\right)+\ln \left(x-3\right)=\ln \left(7x\right)\)
48. \(\ln \left(x\right)+\ln \left(x-6\right)=\ln \left(6x\right)\)
- Contestar
-
1. \(\text{log}_3 (4)\)
3. \(\text{log}_3 (7)\)
5. \(\text{log}_3 (5)\)
7. \(\text{log}_7 (2)\)
9. \(\text{log} (6x^9)\)
11. \(\text{ln} (2x^7)\)
13. \(\text{log}(x^2 (x + 1)^3)\)
15. \(\text{log} (\dfrac{xz^3}{\sqrt{y}})\)
17. \(15\text{log}(x) + 13 \text{log}(y) - 19 \text{log}(z)\)
19. \(-2\text{ln} (a) + 4\text{ln}(b) - 5 \text{ln}(c)\)
21. \(\dfrac{3}{2} \text{log}(x) - 2 \text{log}(y)\)
23. \(\text{ln}(y) + \dfrac{1}{2} (\text{ln} (y) - \text{ln} (1 - y))\)
25. \(\dfrac{8}{3} \text{log} (x) + \dfrac{14}{3} \text{log} (y)\)
27. \(x \approx -0.717\)
29. \(x \approx -6.395\)
31. \(t \approx 17.329\)
33. \(x = \dfrac{2}{7}\)
35. \(x \approx 0.123\)
37. \(x \approx 4.642\)
39. \(x \approx 30.158\)
41. \(x \approx -2.889\)
43. \(x \approx 6.873\)o\(x \approx -0.873\)
45. \(x = \dfrac{12}{11} \approx 1.091\)
47. \(x = 10\)