Logarithms are useful when the measurements in a dataset have a wide range of values, all greater than zero. Positive values are needed because only positive numbers have logarithms, since no exponent of 10 exists that gives a negative or zero result. Use of a negative value in the LOG10 spreadsheet function, or with the LOG key on a calculator, will give an error message.
The logarithm values themselves can be zero, in which case the original value is exactly 1, or negative, in which case the original value is less than 1 (e.g., -2 is the logarithm of 0.01 = 10-2).
Example 1: For each of the values listed below, use reasoning to answer these two questions: [i] Does the value have a logarithm greater than 0? [ii] Does the value have a logarithm that is a whole number?
[a] 582 [b] 10,000 [c] 0.23 [d] -48 [e] 6.2×1026 [f] 493.57285 [g] 0.001
The whole-number powers of 10 are obvious: 101 = 10, 102 = 100, 103 = 1000, etc., as are the whole-number negative powers: 10-1 = 0.1, 10-2 = 0.01, 10-3 = 0.001, etc. The logarithm of any of these numbers is simply the corresponding exponent of 10. Also, 100 = 1 (any value to a zero power equals one), so numbers greater than 1 have positive logarithms and numbers less than one have negative logarithms.
[a] The logarithm of 582 is positive and is not an integer.
[b] The logarithm of 10,000 is positive and is an integer.
[c] The logarithm of 0.23 is negative and is not an integer.
[d] This number -48 is not positive, and therefore it does not have a logarithm.
[e] The logarithm of 6.2×1026 is positive and is not an integer.
[f] The logarithm of 493.57285 is positive and is not an integer.
[g] The logarithm of 0.001 is negative and is an integer.
Example 2: For each of these logarithms, which two integer powers of 10 is the original value between?
[a] 1.634 [b] 4.195 [c] -2.593 [d] -0.345 [e] 0.683
[a] 101 = 10 and 102 = 100 (since 1.634 is between 1 and 2).
[b] 104 = 10,000 and 105 = 100,000 (since 4.195 is between 4 and 5).
[c] 10-2 = 0.01 and 10-3 = 0.001 (since -2.593 is between -2 and -3).
[d] 100 = 1 and 10-1 = 0.1 (since -0.345 is between 0 and -1).
[e] 100 = 1 and 101 = 10 (since 0.683 is between 0 and 1).
Finding logarithms of values, or values from their logarithms
In spreadsheets, we compute the base-10 logarithm of a number with the LOG10 function. Thus a cell containing the formula “=LOG10(3100)” will display the result 3.491361694. Note that logarithms, like trigonometric functions, almost always give values that are non-repeating decimals, so the logarithm values used are approximate rather than exact. The exception is for whole-number powers of the base, so that the base-10 logarithm of 1,000,000 is exactly 6, and that of 0.001 is exactly –3.
Since a logarithm is an exponent, you can always get back the original value by using the logarithm value as an exponent for the base. Thus 103.491 (the formula “=10^3.491” in a spreadsheet) will evaluate to nearly 3100, although there will a small difference due to rounding-error propagation because 3.491 is a rounded-off version of the logarithm. Most calculators have a LOG key that has the same effect as the LOG10 spreadsheet function. The inverse function for a calculator’s LOG key is 10x, which coverts a base-10 logarithm back to the original value.
Example 3: Find the common logarithms of these numbers, to three decimal places:
[a] 48,300 [b] 2 [c] 0.055 [d] 7.2 [e] 2.6×1013 [f] 1.9×10-5
Solution approaches (you can use either one):
[i] With a calculator, enter the value, then press the LOG key to see the logarithm. Use the EE or EXP key to enter the exponent of the numbers stated in scientific notation.
[ii] In a worksheet, enter a LOG10 formula with the value in parentheses, such as “=LOG10(7.2)”. The numbers in scientific notation can be entered in “E format” – in the case as “2.5E13” and “1.9E-5”; you may wish to use the Format > Cells option to covert the format of result to Number or General.
[a] 4.684 [b] 0.301 [c] –1.260 [d] 0.857 [e] 13.415 [f] –4.721
Example 4: Find the values, to three significant digits, which have these numbers as logarithms:
[a] 2.321 [b] –2.763 [c] 0.632 [d] –12.485 [e] 5.364 [f]26.931
Solution approaches (also remember to round to three significant digits):
[i] With a calculator, enter the logarithm, then press the 10x key to see the value.
[ii] In a worksheet, enter a formula with the logarithm as the exponent of 10, such as “=10^2.321”.
Answers: [a] 209 [b] 0.00173 [c] 4.29 [d] 3.27×10-13 [e] 23,100 [f] 8.53×1026