Log Table: How to Read & Use a Logarithm Table
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Log Table: How to Read & Use a Logarithm Table
TL;DR
A log table is a reference chart that gives the base-10 logarithm of a number, used to multiply, divide, and find powers by hand. To read a value, you split the logarithm into two parts: the characteristic (the integer part) and the mantissa (the decimal part, read from the table).
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Bhanzu Team Last updated on June 29, 20264 min read
The Log Table (base 10, common values)
This is the block to keep. It lists the common logarithm of the whole numbers 1 to 10.
Table 1: Common logarithms (base 10) of 1 to 10.
| Number nnn | log10n\log_{10} nlog10n |
|---|---|
| 1 | 0.0000 |
| 2 | 0.3010 |
| 3 | 0.4771 |
| 4 | 0.6021 |
| 5 | 0.6990 |
| 6 | 0.7782 |
| 7 | 0.8451 |
| 8 | 0.9031 |
| 9 | 0.9542 |
| 10 | 1.0000 |
A full printed log table extends this to four-digit numbers using rows, columns, and a mean-difference section.
Characteristic And Mantissa
Every common logarithm has two parts, separated by the decimal point.
Characteristic: the integer part. For a number greater than 1, it is one less than the number of digits before the decimal point.
Mantissa: the decimal part, always positive, read straight from the log table.
For log234\log 234log234, the number has 3 digits, so the characteristic is 3−1=23 - 1 = 23−1=2. The mantissa for the digits "234" is read from the table as 0.36920.36920.3692, giving log234≈2.3692\log 234 \approx 2.3692log234≈2.3692.
For a number less than 1, the characteristic is negative. log0.234\log 0.234log0.234 has characteristic −1-1−1, written in bar notation as 1ˉ.3692\bar{1}.36921ˉ.3692, where only the characteristic is negative and the mantissa stays positive.
How To Read A Log Table
Use the RCM order: Row, Column, Mean difference.
Ignore the decimal point and take the significant digits of the number.
Row: find the row for the first two digits.
Column: move across to the column for the third digit, and read the four-figure value.
Mean difference: add the mean-difference value for the fourth digit.
Place the decimal: prefix the characteristic, counting digits before the original decimal point.
For log18.25\log 18.25log18.25:
Row 18, column 2 gives 0.26130.26130.2613.
Mean difference for 5 adds 121212 (i.e. 0.00120.00120.0012), giving mantissa 0.26250.26250.2625.
The number 18.25 has 2 digits before the decimal, so the characteristic is 2−1=12 - 1 = 12−1=1.
log18.25≈1.2625\log 18.25 \approx 1.2625log18.25≈1.2625.
Using The Antilog Table
An antilog table reverses the process: it turns a logarithm back into the original number.
To find the antilog of 2.36922.36922.3692:
Use the mantissa 0.36920.36920.3692 in the antilog table (row .36.36.36, column 999, plus mean difference for 222) to get the digits, about 234023402340.
The characteristic is 222, so multiply by 10210^{2}102, placing the decimal after 1+2=31 + 2 = 31+2=3 digits.
Antilog 2.3692≈2342.3692 \approx 2342.3692≈234.
The log and antilog tables are a matched pair: log goes number → logarithm, antilog goes logarithm → number.
Where The Log Table Appears
Log tables were the standard calculating tool from the 1600s until handheld calculators arrived in the 1970s. They reduced multiplication and division to addition and subtraction, which is far faster by hand.
Navigation and astronomy: sailors and astronomers multiplied large numbers using log tables for centuries.
Engineering and the slide rule: the slide rule is a log table built into a physical instrument.
Exams: many board and competitive exams still permit a printed log table instead of a calculator.
Understanding logarithms: reading a table makes the properties of logarithms concrete, since multiplication of numbers becomes addition of their logs.
Common Mistakes With A Log Table
1. Miscounting the characteristic
The characteristic for a number greater than 1 is the digit count minus one, not the digit count, so for log234\log 234log234 it is 222, not 333. Counting it as the number of digits is the most frequent slip.
2. Making the mantissa negative
The mantissa is always positive, even when the whole logarithm is negative. For numbers below 1, keep the mantissa positive and put the negative sign only on the characteristic, using bar notation such as 1ˉ.3692\bar{1}.36921ˉ.3692.
3. Reading the wrong row or column
The row uses the first two significant digits and the column uses the third digit. Skipping the mean-difference step for the fourth digit gives a value that is close but wrong in the last place.
Practice Questions On The Log Table
Use the common-log table above, then check your answers below.
Write the value of log105\log_{10} 5log105.
Write the value of log107\log_{10} 7log107.
What is the characteristic of log4567\log 4567log4567?
What is the characteristic of log0.0456\log 0.0456log0.0456?
State log8\log 8log8 in full (characteristic and mantissa).
Answers
log105≈0.6990\log_{10} 5 \approx 0.6990log105≈0.6990.
log107≈0.8451\log_{10} 7 \approx 0.8451log107≈0.8451.
The number has 4 digits, so the characteristic is 4−1=34 - 1 = 34−1=3.
The first significant digit sits in the second decimal place, so the characteristic is −2-2−2, written 2ˉ\bar{2}2ˉ in bar notation.
log8≈0.9031\log 8 \approx 0.9031log8≈0.9031.