Log, ln and Mathematical Operations

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)