Log Table (Base 10, Natural Log and Base 2)

A log table gives you the decimal logarithm (base 10) for any number from 1.000 to 9.999 to four decimal places. For numbers outside that range, you add or subtract the characteristic based on the number of digits. The most common mistake is thinking log(x) and ln(x) are interchangeable; they are different functions with different bases (10 and e, respectively). Here are the log₁₀ values for 1 through 100, followed by natural log and base-2 values, all to four decimal places.

For base 10, log(1) = 0.0000, log(10) = 1.0000, log(100) = 2.0000. For natural logs, ln(1) = 0.0000, ln(10) ≈ 2.3026, ln(100) ≈ 4.6052. For base 2, log₂(1) = 0.0000, log₂(10) ≈ 3.3219, log₂(100) ≈ 6.6439. Use this table to look up or print log values.

Understanding the Parts

A four-figure log table format gives you the mantissa (the decimal part) for numbers 1.000 to 9.999. The mantissa is always between 0.0000 and 0.9999. Rows are indexed by the first three digits of the number (100 to 999), columns by the fourth digit (0 to 9). A separate mean differences table allows you to adjust for a fifth digit.

The integer part is the characteristic. For a number greater than or equal to 1, the characteristic equals the number of digits to the left of the decimal point minus 1. For 100, that is 3 minus 1 = 2, so log₁₀(100) = 2. For 1, it is 1 minus 1 = 0, so log₁₀(1) = 0. For 0.5, which has no digits to the left and one leading zero after the decimal, the characteristic is, 2.

Worked Example and Common Pitfall

Look up 3.142 in the table: find row 314, column 2, read mantissa (e.g., 0.4969), then add the mean difference for digit 2 from the mean differences table (say 3), giving 0.4972. The characteristic for 3.142 is 0 (one digit left of decimal), so log₁₀(3.142) ≈ 0.4972.

The failure case: getting the characteristic wrong when the number is between 0 and 1. For 0.003, characteristic is, 3 (three leading zeros after decimal plus one), not 0. Linear interpolation between tabulated 4-digit mantissas using the mean differences table gives about 2-3 decimal places of accuracy.

Before calculators, multiplication was turned into addition using a log tables. To multiply 3.142 by 5.678, look up log₁₀(3.142) as above (0.Add them: 0.4972 + 0.7537 = 1.2509. The sum is the log of the product. Now find the antilog of 1.2509: look up mantissa 0.2509 in the log table backwards (find what number gives that mantissa, say 1.78), then apply the characteristic 1, giving 17.8. The product is 17.8. This method relies on the product rule: log_b(MN) = log_b(M) + log_b(N).

The failure case: misapplying the product rule on log(a+b) as if it equals log(a) + log(b). It does not. Check your result by estimating: 3.142 × 5.678 is roughly 3 × 6 = 18, so 17.8 is plausible.

The antilog is the inverse operation. For a common logarithm, antilog₁₀(mantissa) = 10^mantissa. For a natural log, antilog_e(y) = e^y. To find the number from its log, you reverse the table lookup process.

Suppose you have a log value of 1.4972. The characteristic is 1, the mantissa is 0.4972. Look up mantissa 0.4972 in the log table: find the row where the mantissa is closest. For example, mantissa 0.4972 corresponds to number 3.142 (since log₁₀(3.142) ≈ 0.4972). Multiply by 10^1 = 10, giving 31.42. The antilog of 1.4972 in base 10 is 31.42.

The failure case: trying to find antilog of a negative log without separating characteristic and mantissa. For log =, 0.3010, write as, 1 + 0.6990 (characteristic, 1, mantissa 0.6990). Look up mantissa 0.6990 (which corresponds to 5), then multiply by 10^, 1 = 0.1, giving 0.5. Not, 0.3010 directly.

Memorise the Key Values

The log 1 to 10 values are the most frequently used: log₁₀(2) ≈ 0.3010, log₁₀(3) ≈ 0.4771, log₁₀(4) ≈ 0.6021, log₁₀(5) ≈ 0.6990, log₁₀(6) ≈ 0.7782, log₁₀(7) ≈ 0.8451, log₁₀(8) ≈ 0.9031, log₁₀(9) ≈ 0.9542, log₁₀(10) = 1.0000. Memorise these; they let you estimate any log mentally using the product and power rules.

Applications in Chemistry, Seismology, and Acoustics

In chemistry, pH is defined as, log₁₀([H⁺]), so a hydrogen ion concentration [H⁺] = 1.0 × 10⁻⁴ mol/L gives pH = 4.00. The Richter scale uses log₁₀(amplitude / reference amplitude); each whole-number step multiplies amplitude by 10 and energy by about 31.6 (10^1.5). The decibel (dB) uses 10 log₁₀(power ratio).

Use in Computer Science

In computer science, a log table 1 to 100 in base 2 helps analyze algorithm complexity: log₂(100) ≈ 6.64 means a binary search on 100 items takes at most 7 steps. The change of base formula, log_b(x) = log_a(x) / log_a(b), lets you convert between bases using any known log.

For a printable log table, list numbers 1 to 100 in a grid with columns for log₁₀, ln, and log₂. Use a monospace font and 4 decimal places. Keep the characteristic and mantissa separated by a decimal point, with the mantissa always 4 digits. Print at least 10 rows per page for legibility. A good layout is a single column per page, or two columns side by side for 1-50 and 51-100.

The characteristic rule is printed at the top: for number ≥ 1, characteristic = digit count, 1; for number < 1, characteristic =, (leading zeros after decimal + 1). The domain is x > 0; log of zero or a negative number is undefined in real numbers.