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:

Column N shows the integer base from $10$ to $99$
Columns $0$ through $9$ give the mantissa for the first decimal digit
Columns Mean Diff 1 through Mean Diff 9 give the correction for the second decimal digit
The calculator above the table accepts any positive value and returns the four-decimal log value
The column header row and the input area stay sticky as you scroll

Using the Calculate Input

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

Behavior:

The input accepts decimals; values are rounded to two decimal places for the lookup
The result appears as a four-decimal mantissa just above the table
The matching row scrolls into view and centers automatically
The cell used for the first decimal and the mean-difference cell used for the second decimal both highlight in yellow
Negative numbers, zero, and non-numeric input show an error message in place of a result
Click 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:

Find the row for the integer part of your number (for $25.43$ , use row $25$ )
Read the value in the column matching the first decimal (column $4$ )
Read the value in Mean Diff matching the second decimal (Mean Diff $3$ )
Add the two values to get your final mantissa

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:

Main grid cells show the original number that the mantissa corresponds to, along with the four-decimal value
Mean-difference cells show the differential value itself

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

The row containing the integer base
The cell at the intersection with the first-decimal column
The mean-difference cell at the corresponding second-decimal column

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:

Locate row $23$
Read column $4$ — that gives the mantissa for $23.4$
Read Mean Diff 7 on the same row — that adds the contribution from the second decimal
Sum the two values to get the mantissa for $23.47$

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:

The characteristic is the integer part, determined by the magnitude of the original number
The 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.

Logarithm Table