What is a logarithm table?

A logarithm table lists the common (base-10) logarithm values of numbers within a given range, typically printed to four decimal places. This table covers every value from 10 to 99.99, organized so each row gives 19 lookup values for one integer base: 10 main mantissa columns plus 9 mean-difference corrections.

How do I read a logarithm table?

Find the row matching the integer part of your number, read the column matching the first decimal digit, then add the mean-difference value from the column matching the second decimal digit. For log of 23.47, you use row 23, column 4 for the main mantissa, and Mean Diff 7 for the second-decimal correction, summing the two.

What is the difference between common log and natural log?

The common logarithm uses base 10 and is often written as log. The natural logarithm uses base e (about 2.71828) and is written as ln. To convert a common log to a natural log, multiply by ln(10), which is approximately 2.302585.

What are the characteristic and mantissa?

Every common logarithm splits into an integer characteristic and a fractional mantissa. The mantissa is what you read from the table. The characteristic is the integer part, determined by the magnitude of the number: a number with d digits before the decimal point has characteristic d minus 1. So 347, 34.7, and 3.47 all share the same mantissa but have characteristics 2, 1, and 0.

Logarithm Table - Base 10 Log Values

Getting Started with the Table

The table covers common (base-10) logarithms for values from

10

to

99.99

, organized as a four-decimal mantissa table. To find a log, you read two table sections: the main grid for the first decimal digit, then a mean-difference column for the second.

Quick orientation:

N shows the integer base from $10$ to $99$
$0$ through $9$ give the mantissa for the first decimal digit
Mean Diff $1$ through Mean Diff $9$ give the correction for the second decimal digit

Using the Calculate Input

Type any positive number into the input and click Calculate to compute its logarithm.

Behavior:

Reset to clear the input, the result, and all highlights

Reading the Mean Differences

The mean-difference columns extend the lookup beyond one-decimal precision. Without them, the table would only handle values like

25.0

,

25.1

,

25.2

; the mean differences let you read

\log(25.43)

or

\log(25.78)

directly.

How to use them:

$25.43$ , use row $25$ )
$4$ )
Mean Diff matching the second decimal (Mean Diff $3$ )

Mean differences shrink as the row number grows because the spacing between consecutive log values gets smaller for larger numbers — this is the curvature of the log function shown in the table itself.

Hover Tooltips and Cell Detail

Every cell in the table has a tooltip on hover:

After a calculation, the table highlights three things at once:

The sticky column header stays visible while you scroll through the

90

-row table, so the digit labels at the top remain readable at any vertical position.

Looking Up a Value by Hand

For

\log_{10}(23.47)

using the table alone:

$23$
$4$ — that gives the mantissa for $23.4$
Mean Diff $7$ on the same row — that adds the contribution from the second decimal
$23.47$

The result is the mantissa only. The characteristic (integer part of the log) is determined by the number of digits before the decimal point in your original value, not from the table. For

23.47

the characteristic is

1

because there are two digits before the decimal point, so

\log_{10}(23.47) \approx 1 + \text{(mantissa)}

.

For a deeper look at the lookup procedure, see the logarithms theory page.

What Is a Logarithm?

A logarithm answers the question: to what power must the base be raised to produce a given number? For base

10

:

\log_{10}(x) = y \quad \iff \quad 10^y = x

Simple examples:

\log_{10}(100) = 2

because

10^2 = 100

, and

\log_{10}(1000) = 3

because

10^3 = 1000

.

For values between powers of

10

the logarithm is a non-integer. This table gives those non-integer values to four decimal places for every value from

10

to

99.99

.

For a fuller treatment, see the logarithms page.

Common vs Natural Logarithms

Two logarithm bases dominate practical use:

Common logarithm ( $\log_{10}$ , often written just $\log$ ) uses base $10$ — this table is a common log table
Natural logarithm ( $\ln$ ) uses base $e \approx 2.71828$

Conversion between them uses the change-of-base formula:

\ln(x) = \log_{10}(x) \cdot \ln(10)

with

\ln(10) \approx 2.302585

. So any value read from this table can be converted to a natural log by multiplying by

2.302585

.

For more on this, see the logarithm properties page.

Characteristic and Mantissa

Every common logarithm splits into two parts:

characteristic is the integer part, determined by the magnitude of the original number
mantissa is the fractional part, looked up in the table

For a number with

d

digits before the decimal point, the characteristic is

d - 1

:

$\log_{10}(347) = 2 + \text{mantissa}$
$\log_{10}(34.7) = 1 + \text{mantissa}$
$\log_{10}(3.47) = 0 + \text{mantissa}$

The mantissa stays the same when you shift the decimal point — only the characteristic changes. This is why a single table covering

10

to

99.99

extends to every positive number: pick up the mantissa from the table, then prepend the characteristic determined by counting digits.

Related Tools and Concepts

Related Tools:

Logarithm Calculator - Computes

\log_{10}

and

\ln

directly for any positive input without a lookup.

Antilogarithm Table - The inverse lookup: given a logarithm, find the original number.

Related Concepts:

Logarithm Rules - Product, quotient, and power rules for combining logarithms.

Change of Base - Convert a logarithm in any base into another base.

Exponential Functions - Inverses of logarithms;

\log_b

inverts

b^x

.

Slide Rule - The mechanical computer built on logarithm tables, superseded by electronic calculators.