Log Table: How to Read & Use a Logarithm Table

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.

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 $\log 234$, the number has 3 digits, so the characteristic is $3 - 1 = 2$. The mantissa for the digits "234" is read from the table as $0.3692$, giving $\log 234 \approx 2.3692$.

For a number less than 1, the characteristic is negative. $\log 0.234$ has characteristic $-1$, written in bar notation as $\bar{1}.3692$, 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.

1. Ignore the decimal point and take the significant digits of the number.

2. Row: find the row for the first two digits.

3. Column: move across to the column for the third digit, and read the four-figure value.

4. Mean difference: add the mean-difference value for the fourth digit.

5. Place the decimal: prefix the characteristic, counting digits before the original decimal point.

For $\log 18.25$:

Row 18, column 2 gives $0.2613$.

Mean difference for 5 adds $12$ (i.e. $0.0012$), giving mantissa $0.2625$.

The number 18.25 has 2 digits before the decimal, so the characteristic is $2 - 1 = 1$.

$\log 18.25 \approx 1.2625$.

Using The Antilog Table

An antilog table reverses the process: it turns a logarithm back into the original number.

To find the antilog of $2.3692$:

• Use the mantissa $0.3692$ in the antilog table (row $.36$, column $9$, plus mean difference for $2$) to get the digits, about $2340$.

• The characteristic is $2$, so multiply by $10^{2}$, placing the decimal after $1 + 2 = 3$ digits.

• Antilog $2.3692 \approx 234$.

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 $\log 234$ it is $2$, not $3$. 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 $\bar{1}.3692$.

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.

1. Write the value of $\log_{10} 5$.

2. Write the value of $\log_{10} 7$.

3. What is the characteristic of $\log 4567$?

4. What is the characteristic of $\log 0.0456$?

5. State $\log 8$ in full (characteristic and mantissa).

Answers

1. $\log_{10} 5 \approx 0.6990$.

2. $\log_{10} 7 \approx 0.8451$.

3. The number has 4 digits, so the characteristic is $4 - 1 = 3$.

4. The first significant digit sits in the second decimal place, so the characteristic is $-2$, written $\bar{2}$ in bar notation.

5. $\log 8 \approx 0.9031$, with characteristic $0$ and mantissa $0.9031$.