Ejercicio 299

octubre 30, 2019abril 16, 2019 por Gestor Baldor

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CAPITULO XXXVIII

Logaritmos

Ejercicio 299

Hallar por log el valor de las expresiones siguientes:

$\begin{matrix} \frac{515 \times 78.19}{6.13} \\ \frac{515 \times 78.19}{6.13} & = l o g (\frac{515 \times 78.19}{6.13}) \\ = l o g (515 \times 78.19) - l o g 6.13 \\ = l o g 515 + l o g 78.19 - 0.787460 \\ = l o g 515 + l o g 78.19 - 0.787460 \\ = 2.711807 + 1.893151 - 0.787460 \\ = 3.817497968 \\ = a n t i l o g (3.817497968) \\ = 6568.980424 \end{matrix}$
$\begin{matrix} \frac{23.054 \times 934.5}{8164} \\ \frac{23.054 \times 934.5}{8164} & = l o g (\frac{23.054 \times 934.5}{8164}) \\ = l o g (23.054 \times 934.5) - l o g (8164) \\ = l o g 23.054 + l o g 934.5 - 3.911902996 \\ = 1.362746289 + 2.970579306 - 3.911902996 \\ = 0.421422598 \\ = a n t i l o g (0.421422598) \\ = 2.638897967 \end{matrix}$
$\begin{matrix} \frac{8.14 \times 9.73}{0.6 \times 7.8} \\ \frac{8.14 \times 9.73}{0.6 \times 7.8} & = l o g (\frac{8.14 \times 9.73}{0.6 \times 7.8}) \\ = l o g (8.14 \times 9.73) - l o g (0.6 \times 7.8) \\ = l o g 8.14 + l o g 9.73 - (l o g 0.6 + l o g 7.8) \\ = 0.9106244049 + 0.9881128403 - (- 0.2218487496 + 0.8920946027) \\ = 1.228491392 \\ = a n t i l o g (1.228491392) \\ = 16.92354701 \end{matrix}$
$\begin{matrix} \frac{513.4 \times 9.132}{85.3 \times 10.764} \\ \frac{513.4 \times 9.132}{85.3 \times 10.764} & = l o g (\frac{513.4 \times 9.132}{85.3 \times 10.764}) \\ = l o g (513.4 \times 9.132) - l o g (85.3 \times 10.764) \\ = l o g 513.4 + l o g 9.132 - (l o g 85.3 + l o g 10.764) \\ = 2.710455864 + 0.9605659028 - (1.930949031 + 1.031973689) \\ = 0.7080990469 \\ = a n t i l o g (0.7080990469) \\ = 5.10621441 \end{matrix}$
$\begin{matrix} \frac{53.245 \times 4325.6}{32.815 \times 91.79} \\ \frac{53.245 \times 4325.6}{32.815 \times 91.79} & = l o g (\frac{53.245 \times 4325.6}{32.815 \times 91.79}) \\ = l o g (53.245 \times 4325.6) - l o g (32.815 \times 91.79) \\ = l o g 53.245 + l o g 4325.6 - (l o g 32.815 + l o g 91.79) \\ = 1.726278831 + 3.636046357 - (1.516072409 + 1.96279537) \\ = 1.883457409 \\ = a n t i l o g (1.883457409) \\ = 76.46406978 \end{matrix}$
$\begin{matrix} \frac{32.6 \times (- 841.9)}{0.017 \times 732.14} \\ M u l t i p l i c o p o r - 1 \\ \frac{32.6 \times 841.9}{0.017 \times 732.14} & = l o g (\frac{32.6 \times 841.9}{0.017 \times 732.14}) \\ = l o g (32.6 \times 841.9) - l o g (0.017 \times 732.14) \\ = l o g 32.6 + l o g 841.9 - (l o g 0.017 + l o g 732.14) \\ = 1.5132176 + 2.92526051 - (- 1.769551079 + 2.864594135) \\ = 3.343435053 \\ = a n t i l o g (3.343435053) \\ = 2205.134344 \\ \frac{32.6 \times (- 841.9)}{0.017 \times 732.14} & = - 2205.134344 \end{matrix}$
$\begin{matrix} \frac{95.36 \times (- 0.14)}{(- 83.7) \times (2.936)} \\ \frac{95.36 \times 0.14}{83.7 \times 2.936} & = l o g (\frac{95.36 \times 0.14}{83.7 \times 2.936}) \\ = l o g (95.36 \times 0.14) - l o g (83.7 \times 2.936) \\ = l o g 95.36 + l o g 0.14 - (l o g 83.7 + l o g 2.936) \\ = 1.979366242 - 0.8538719643 - (1.922725458 + 0.4677560512) \\ = - 1.264987231 \\ = a n t i l o g (- 1.264987231) \\ = 0.0543266304 \end{matrix}$
$\begin{matrix} \frac{(- 7.2) \times (- 8.135)}{(- 0.003) \times (9134.7)} & = - \frac{7.2 \times 8.135}{0.003 \times 9134.7} \\ M u l t i p l i c o p o r - 1 \\ \frac{7.2 \times 8.135}{0.003 \times 9134.7} & = l o g (\frac{7.2 \times 8.135}{0.003 \times 9134.7}) \\ = l o g (7.2 \times 8.135) - l o g (0.003 \times 9134.7) \\ = l o g 7.2 + l o g 8.135 - (l o g 0.003 + l o g 9134.7) \\ = 0.8573324964 + 0.9103575573 - (- 2.522878745 + 3.960694289) \\ = 0.3298745101 \\ = a n t i l o g (0.3298745101) \\ = 2.137344412 \\ \frac{(- 7.2) \times (- 8.135)}{(- 0.003) \times (9134.7)} & = - 2.137344412 \end{matrix}$
$\begin{matrix} 3^{5} \times 0. 2^{4} \\ 3^{5} \times 0. 2^{4} & = l o g (3^{5} \times 0. 2^{4}) \\ = l o g 3^{5} + l o g 0. 2^{4} \\ = 5 l o g 3 + 4 l o g 0.2 \\ = 5 (0.4771212547) + 4 (- 0.6989700043) \\ = 2.385606274 - 2.795880017 \\ = - 0.4102737437 \\ = a n t i l o g (- 0.4102737437) \\ = 0.3888 \end{matrix}$
$\begin{matrix} 5^{\frac{1}{2}} \times 3^{\frac{2}{3}} \\ 5^{\frac{1}{2}} \times 3^{\frac{2}{3}} & = l o g (5^{\frac{1}{2}} \times 3^{\frac{2}{3}}) \\ = l o g 5^{\frac{1}{2}} + l o g 3^{\frac{2}{3}} \\ = \frac{1}{2} l o g 5 + \frac{2}{3} l o g 3 \\ = \frac{1}{2} (0.6989700043) + \frac{2}{3} (0.4771212547) \\ = 0.3494850022 + 0.3180808365 \\ = 0.6675658386 \\ = a n t i l o g (0.6675658386) \\ = 4.651208827 \end{matrix}$
$\begin{matrix} 2^{\frac{1}{5}} \times 3^{\frac{1}{2}} \times 5^{\frac{3}{4}} \\ 2^{\frac{1}{5}} \times 3^{\frac{1}{2}} \times 5^{\frac{3}{4}} & = l o g (2^{\frac{1}{5}} \times 3^{\frac{1}{2}} \times 5^{\frac{3}{4}}) \\ = l o g 2^{\frac{1}{5}} + l o g 3^{\frac{1}{2}} + l o g 5^{\frac{3}{4}} \\ = \frac{1}{5} l o g 2 + \frac{1}{2} l o g 3 + \frac{3}{4} l o g 5 \\ = \frac{1}{5} (0.3010299957) + \frac{1}{2} (0.4771212547) + \frac{3}{4} (0.6989700043) \\ = 0.06020599913 + 0.2385606274 + 0.5242275033 \\ = 0.8229941297 \\ = a n t i l o g (0.8229941297) \\ = 6.652641639 \end{matrix}$
$\begin{matrix} \frac{3^{8}}{5. 6^{5}} \\ \frac{3^{8}}{5. 6^{5}} & = l o g (\frac{3^{8}}{5. 6^{5}}) \\ = l o g {(3)}^{8} - l o g {(5.6)}^{5} \\ = 8 l o g 3 - 5 l o g 5.6 \\ = 3.816970038 - 3.740940135 \\ = 0.07602990273 \\ = a n t i l o g (0.07602990273) \\ = 1.191324032 \end{matrix}$
$\begin{matrix} \frac{0.5 3^{7}}{2. 5^{3}} \\ \frac{0.5 3^{7}}{2. 5^{3}} & = l o g (\frac{0.5 3^{7}}{2. 5^{3}}) \\ = l o g {(0.53)}^{7} - l o g {(2.5)}^{3} \\ = 7 l o g (0.53) - 3 l o g (2.5) \\ = - 1.930068913 - 1.193820026 \\ = - 3.123888939 \\ = a n t i l o g (- 3.123888939) \\ = 7.518151 \times 1 0^{- 4} \end{matrix}$
$\begin{matrix} \frac{3^{\frac{2}{5}}}{2^{\frac{5}{3}}} \\ \frac{3^{\frac{2}{5}}}{2^{\frac{5}{3}}} & = l o g (\frac{3^{\frac{2}{5}}}{2^{\frac{5}{3}}}) \\ = l o g 3^{\frac{2}{5}} - l o g 2^{\frac{5}{3}} \\ = \frac{2}{5} l o g 3 - \frac{5}{3} l o g 2 \\ = 0.1908485019 - 0.5017166594 \\ = - 0.3108681576 \\ = a n t i l o g (- 0.3108681576) \\ = 0.4888007262 \end{matrix}$
$\begin{matrix} \sqrt{7.86 \times 8.14} \\ {(7.86 \times 8.14)}^{\frac{1}{2}} & = l o g {(7.86 \times 8.14)}^{\frac{1}{2}} \\ = \frac{1}{2} l o g (7.86 \times 8.14) \\ = \frac{1}{2} (l o g 7.86 + l o g 8.14) \\ = \frac{1}{2} (0.895422546 + 0.9106244049) \\ = 0.9030234755 \\ = a n t i l o g (0.9030234755) \\ = 7.998774906 \end{matrix}$
$\begin{matrix} \sqrt{932.5 \times 813.6 \times 0.005} \\ {(932.5 \times 813.6 \times 0.005)}^{\frac{1}{2}} & = l o g {(932.5 \times 813.6 \times 0.005)}^{\frac{1}{2}} \\ = \frac{1}{2} l o g (932.5 \times 813.6 \times 0.005) \\ = \frac{1}{2} (l o g 932.5 + l o g 813.6 + l o g 0.005) \\ = \frac{1}{2} (2.96964884 + 2.91041094 - 2.301029996) \\ = 1.789514892 \\ = a n t i l o g (1.789514892) \\ = 61.59066488 \end{matrix}$
$\begin{matrix} \sqrt{\frac{93.7 \times 104.2}{8.35 \times 7.3}} \\ {(\frac{93.7 \times 104.2}{8.35 \times 7.3})}^{\frac{1}{2}} & = l o g {(\frac{93.7 \times 104.2}{8.35 \times 7.3})}^{\frac{1}{2}} \\ = \frac{1}{2} l o g (\frac{93.7 \times 104.2}{8.35 \times 7.3}) \\ = \frac{1}{2} [l o g (93.7 \times 104.2) - l o g (8.35 \times 7.3)] \\ = \frac{1}{2} [l o g 93.7 + l o g 104.2 - (l o g 8.35 + l o g 7.3)] \\ = \frac{1}{2} [1.971739591 + 2.017867719 - (0.9216864755 + 0.8633228601)] \\ = 1.102298987 \\ = a n t i l o g (1.102298987) \\ = 12.65607347 \end{matrix}$
$\begin{matrix} \sqrt[3]{23.725 \times (- 9.182) \times 7.184} & = - \sqrt[3]{23.725 \times (9.182) \times 7.184} \\ M u l t i p l i c o p o r - 1 \\ {(23.725 \times 9.182 \times 7.184)}^{\frac{1}{3}} & = l o g {(23.725 \times 9.182 \times 7.184)}^{\frac{1}{3}} \\ = \frac{1}{3} l o g (23.725 \times 9.182 \times 7.184) \\ = \frac{1}{3} [l o g 23.725 + l o g 9.182 + l o g 7.184] \\ = \frac{1}{3} [1.375206221 + 0.9629372884 + 0.8563663237] \\ = 1.064836611 \\ = a n t i l o g (1.064836611) \\ = 11.61011739 \\ \sqrt[3]{23.725 \times (- 9.182) \times 7.184} & = - 11.61011739 \end{matrix}$
$\begin{matrix} \sqrt[4]{\frac{12316 \times 0.25}{931.8 \times 0.07}} \\ {(\frac{12316 \times 0.25}{931.8 \times 0.07})}^{\frac{1}{4}} & = l o g {(\frac{12316 \times 0.25}{931.8 \times 0.07})}^{\frac{1}{4}} \\ = \frac{1}{4} l o g (\frac{12316 \times 0.25}{931.8 \times 0.07}) \\ = \frac{1}{4} [l o g (12316 \times 0.25) - l o g (931.8 \times 0.07)] \\ = \frac{1}{4} [l o g 12316 + l o g 0.25 - (l o g 931.8 + l o g 0.07)] \\ = \frac{1}{4} [4.09046968 - 0.6020599913 - (2.969322706 - 1.15490196)] \\ = 0.4184972357 \\ = a n t i l o g (0.4184972357) \\ = 2.621182355 \end{matrix}$
$\begin{matrix} \sqrt[5]{\frac{56813}{22117}} \\ {(\frac{56813}{22117})}^{\frac{1}{5}} & = l o g {(\frac{56813}{22117})}^{\frac{1}{5}} \\ = \frac{1}{5} l o g (\frac{56813}{22117}) \\ = \frac{1}{5} [l o g (56813) - l o g (22117)] \\ = \frac{1}{5} [4.754447723 - 4.344726218] \\ = 0.08194430096 \\ = a n t i l o g (0.08194430096) \\ = 1.207658941 \end{matrix}$
$\begin{matrix} {(\frac{0.0316}{0.1615})}^{\frac{3}{2}} \\ {(\frac{0.0316}{0.1615})}^{\frac{3}{2}} & = l o g {(\frac{0.0316}{0.1615})}^{\frac{3}{2}} \\ = \frac{3}{2} l o g (\frac{0.0316}{0.1615}) \\ = \frac{3}{2} [l o g (0.0316) - l o g (0.1615)] \\ = \frac{3}{2} [- 1.500312917 + 0.7918274733] \\ = - 1.062728166 \\ = a n t i l o g (- 1.062728166) \\ = 0.08655094896 \end{matrix}$
$\begin{matrix} \frac{3^{\frac{3}{4}}}{5^{\frac{2}{3}}} \\ \frac{3^{\frac{3}{4}}}{5^{\frac{2}{3}}} & = l o g (\frac{3^{\frac{3}{4}}}{5^{\frac{2}{3}}}) \\ = l o g {(3)}^{\frac{3}{4}} - l o g {(5)}^{\frac{2}{3}} \\ = \frac{3}{4} l o g (3) - \frac{2}{3} l o g (5) \\ = \frac{3}{4} (0.4771212547) - \frac{2}{3} (0.6989700043) \\ = - 0.1081390619 \\ = a n t i l o g (- 0.1081390619) \\ = 0.7795804475 \end{matrix}$
$\begin{matrix} \sqrt[7]{\frac{15}{4}} \\ {(\frac{15}{4})}^{\frac{1}{7}} & = l o g {(\frac{15}{4})}^{\frac{1}{7}} \\ = \frac{1}{7} l o g (\frac{15}{4}) \\ = \frac{1}{7} [l o g (15) - l o g (4)] \\ = \frac{1}{7} [1.176091259 - 0.6020599913] \\ = 0.08200446682 \\ = a n t i l o g (0.08200446682) \\ = 1.207826258 \end{matrix}$
$\begin{matrix} \sqrt[5]{- \frac{5}{3}} & = - \sqrt[5]{\frac{5}{3}} \\ M u l t i p l i c o p o r - 1 \\ {(\frac{5}{3})}^{\frac{1}{5}} & = l o g {(\frac{5}{3})}^{\frac{1}{5}} \\ = \frac{1}{5} l o g (\frac{5}{3}) \\ = \frac{1}{5} [l o g (5) - l o g (3)] \\ = \frac{1}{5} [0.6989700043 - 0.4771212547] \\ = 0.04436974992 \\ = a n t i l o g (0.04436974992) \\ = 1.107566343 \\ \sqrt[5]{- \frac{5}{3}} & = - 1.107566343 \end{matrix}$
$\begin{matrix} {(\frac{5}{8})}^{\frac{6}{5}} \\ {(\frac{5}{8})}^{\frac{6}{5}} & = l o g {(\frac{5}{8})}^{\frac{6}{5}} \\ = \frac{6}{5} l o g (\frac{5}{8}) \\ = \frac{6}{5} [l o g (5) - l o g (8)] \\ = \frac{6}{5} [0.6989700043 - 0.903089987] \\ = - 0.2449439792 \\ = a n t i l o g (- 0.2449439792) \\ = 0.5689263134 \end{matrix}$
$\begin{matrix} \sqrt{\frac{3}{5}} \times \sqrt[3]{\frac{5}{7}} \\ {(\frac{3}{5})}^{\frac{1}{2}} \times {(\frac{5}{7})}^{\frac{1}{3}} & = l o g [{(\frac{3}{5})}^{\frac{1}{2}} \times {(\frac{5}{7})}^{\frac{1}{3}}] \\ = l o g {(\frac{3}{5})}^{\frac{1}{2}} + l o g {(\frac{5}{7})}^{\frac{1}{3}} \\ = \frac{1}{2} l o g (\frac{3}{5}) + \frac{1}{3} l o g (\frac{5}{7}) \\ = \frac{1}{2} [l o g (3) - l o g (5)] + \frac{1}{3} [l o g (5) - l o g (7)] \\ = \frac{1}{2} [0.4771212547 - 0.6989700043] + \frac{1}{3} [0.6989700043 - 0.84509804] \\ = - 0.15963372 \\ = a n t i l o g (- 0.15963372) \\ = 0.6924147009 \end{matrix}$
$\begin{matrix} \sqrt[7]{2} \times \sqrt[5]{3} \times \sqrt[3]{0.2} \\ 2^{\frac{1}{7}} \times 3^{\frac{1}{5}} \times {(0.2)}^{\frac{1}{3}} & = l o g [2^{\frac{1}{7}} \times 3^{\frac{1}{5}} \times {(0.2)}^{\frac{1}{3}}] \\ = l o g {(2)}^{\frac{1}{7}} + l o g {(3)}^{\frac{1}{5}} + l o g {(0.2)}^{\frac{1}{3}} \\ = \frac{1}{7} l o g (2) + \frac{1}{5} l o g (3) + \frac{1}{3} l o g (0.2) \\ = \frac{1}{7} (0.3010299957) + \frac{1}{5} (0.4771212547) + \frac{1}{3} (- 0.6989700043) \\ = - 0.09456146541 \\ = a n t i l o g (- 0.09456146541) \\ = 0.8043379031 \end{matrix}$
$\begin{matrix} \frac{\sqrt{32.14} \times \sqrt[3]{59.3}}{\sqrt[4]{317.6}} \\ \frac{{(32.14)}^{\frac{1}{2}} \times {(59.3)}^{\frac{1}{3}}}{{(317.6)}^{\frac{1}{4}}} & = l o g [\frac{{(32.14)}^{\frac{1}{2}} \times {(59.3)}^{\frac{1}{3}}}{{(317.6)}^{\frac{1}{4}}}] \\ = l o g ({(32.14)}^{\frac{1}{2}} \times {(59.3)}^{\frac{1}{3}}) - l o g {(317.6)}^{\frac{1}{4}} \\ = l o g {(32.14)}^{\frac{1}{2}} + l o g {(59.3)}^{\frac{1}{3}} - \frac{1}{4} l o g (317.6) \\ = \frac{1}{2} l o g (32.14) + \frac{1}{3} l o g (59.3) - \frac{1}{4} (2.501880494) \\ = \frac{1}{2} (1.507045872) + \frac{1}{3} (1.773054693) - \frac{1}{4} (2.501880494) \\ = 0.7190710439 \\ = a n t i l o g (0.7190710439) \\ = 5.236860966 \end{matrix}$
$\begin{matrix} \sqrt{\frac{{(0.75)}^{2} \times 39.162}{0.07 \times 3.89}} \\ {(\frac{{(0.75)}^{2} \times 39.162}{0.07 \times 3.89})}^{\frac{1}{2}} & = l o g {(\frac{{(0.75)}^{2} \times 39.162}{0.07 \times 3.89})}^{\frac{1}{2}} \\ = \frac{1}{2} l o g (\frac{{(0.75)}^{2} \times 39.162}{0.07 \times 3.89}) \\ = \frac{1}{2} [l o g ({(0.75)}^{2} \times 39.162) - l o g (0.07 \times 3.89)] \\ = \frac{1}{2} [l o g {(0.75)}^{2} + l o g (39.162) - (l o g (0.07) + l o g (3.89))] \\ = \frac{1}{2} [2 (- 0.1249387366) + 1.592864863 - (- 1.15490196 + 0.5899496013)] \\ = 0.9539698743 \\ = a n t i l o g (0.9539698743) \\ = 8.994351882 \end{matrix}$
$\begin{matrix} \sqrt{\frac{{(0.2)}^{3} \times {(0.3)}^{2}}{{(0.05)}^{4} \times 3.25}} \\ {(\frac{{(0.2)}^{3} \times {(0.3)}^{2}}{{(0.05)}^{4} \times 3.25})}^{\frac{1}{2}} & = l o g {(\frac{{(0.2)}^{3} \times {(0.3)}^{2}}{{(0.05)}^{4} \times 3.25})}^{\frac{1}{2}} \\ = \frac{1}{2} l o g (\frac{{(0.2)}^{3} \times {(0.3)}^{2}}{{(0.05)}^{4} \times 3.25}) \\ = \frac{1}{2} [l o g ({(0.2)}^{3} \times {(0.3)}^{2}) - l o g ({(0.05)}^{4} \times 3.25)] \\ = \frac{1}{2} [l o g {(0.2)}^{3} + l o g {(0.3)}^{2} - (l o g {(0.05)}^{4} + l o g (3.25))] \\ = \frac{1}{2} [3 l o g (0.2) + 2 l o g (0.3) - (4 l o g (0.05) + l o g (3.25))] \\ = \frac{1}{2} [3 (- 0.6989700043) + 2 (- 0.5228787453) - (4 (- 1.301029996) + 0.511883361)] \\ = 0.7747845591 \\ = a n t i l o g (0.7747845591) \\ = 5.95366726 \end{matrix}$