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## Log, ln and Mathematical Operations

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**Log**• Known as common log or base-10 log • the power to which 10 is raised to equal a certain number • Ex. 103 = 1000; therefore log 1000 = 3 • Ex. Log 0.001 = 1 x 10-3 = -3 • Ex. Log 1 = 0**Significant Figures**• The number of digits after the decimal point equals the number of significant figures in the original number. • Ex. Log 23.5 = 1.371 • 3 sig. fig. = 3 sig. fig. After the decimal pt. • Ex. Log 31.25 = 1.4949 • (4 SF) (5 SF)**Antilogarithms**• Is the reverse of taking a logarithm • For example, log 23.5 = 1.371 • Therefore: antilog 1.371 = 23.5 • Is the process of raising 10 to a particular number to get that number (in this case, 23.5)**Use the following Calculator Keys:**• 10x or SHIFT log or INV log**Sample Exercises**• [1.] Solve for x: • a. Log 1,000,000 = x • b. Log 78.90 = x • c. Log 328.45 = x • [2.] Solve for x: • a. log x = 1.31 • b. 1048.1 = x • c. 103.68 = x**Natural log or Base e logarithm, ln**• the power to which e is raised to equal a certain number • e has a value 2.71828 • Example: e2.303 = 10 or 2.71828(2.303) = 10 • Therefore: ln 10 = 2.303**Natural antilog**• Is the reverse of ln • Use the following Calculator Keys: • ex or INV ln**Sample Exercises**• [1.] Solve for x: • a. ln 5.7 = x • b. ln 321.8 = x • c. ln 7.9 = x • [2.] Solve for x: • a. ln x = 6.78 • b. e-5.156 = x • c. ex = 12.1**Correlation between log and ln**• ln x = 2.303 log x • ln (10) = 2.303 log (10) • = 2.303 (1) • ln and log are correlated by the factor 2.303**Mathematical Operations**• Log xy = log x + log y • Log x/y = log x - log y • log an = n log a • log a 1/n = 1/n log a**Exercise Problems**• Solve the following: • log 4 = • log 0.000001 = • ln 86 • antilog of 77 • natural antilog of 123 • ln(600/740) • log (25 x 200)