Log Calculator

A log calculator finds the logarithm of a positive number in any base, the power you must raise the base to in order to get the number. log₁₀(100) = 2 and log₂(8) = 3.

How to use the log calculator

1. Enter a positive value.
2. Enter the base, such as 10, 2 or 2.718281828 for e.
3. Read the logarithm; exact powers such as log₁₀(1,000) show as whole numbers, with no floating-point error.

How the log calculator works

A logarithm answers the question “what power of the base gives this value?” If b^y = x, then log_b(x) = y. The log calculator works for any positive base other than 1 by using the change-of-base rule:

• log_b(x) = ln(x) ÷ ln(b)
• The value x must be greater than 0
• The base b must be greater than 0 and not equal to 1

Results are shown to 8 decimal places.

Log calculator worked example

log₁₀(100) = 2 because 10² = 100. log₂(8) = 3 because 2³ = 8. For a value that is not an exact power, such as log₁₀(50), the log calculator gives 1.69897000, since 10^1.69897 ≈ 50. For the natural log, enter e ≈ 2.718281828 as the base.

Value	Base	Logarithm
100	10	2
1,000	10	3
50	10	1.69897
0.01	10	−2
8	2	3
1,024	2	10
81	3	4
20	e	2.99573

Logarithm rules

• Product: log(xy) = log x + log y
• Quotient: log(x ÷ y) = log x − log y
• Power: log(xⁿ) = n log x
• log_b(1) = 0 and log_b(b) = 1

These rules, listed in the NIST Digital Library of Mathematical Functions, let you split hard problems into simple ones. For example, log₁₀(5,000) = log₁₀(5) + log₁₀(1,000) = 0.69897 + 3 = 3.69897.

Common bases

Base 10 is the common logarithm, used in pH, decibels and the Richter-style magnitude scales. Base e is the natural logarithm, written ln, which appears in growth, decay and calculus. Base 2 is used in computing: log₂(1,024) = 10 tells you that 1,024 needs 10 bits. The log calculator handles all three, plus any other valid base.

Why negative values fail

No positive base raised to a real power gives zero or a negative number, so the logarithm of 0 or a negative value is undefined in real numbers. Values between 0 and 1 have negative logs: log₁₀(0.01) = −2. If the log calculator shows an error, check that the value is positive and the base is not 1.

Working logs without a calculator

Before electronic calculators, people used printed log tables and slide rules, which turn multiplication into addition. To multiply two numbers you added their logs and looked up the result. You can still estimate logs in your head with a few anchors: log₁₀(2) ≈ 0.301, log₁₀(3) ≈ 0.477 and log₁₀(7) ≈ 0.845. Then log₁₀(6) = 0.301 + 0.477 = 0.778, and log₁₀(200) = 0.301 + 2 = 2.301.

The number of digits in a whole number is one more than the whole-number part of its base-10 log, so a number with a log of 5.4 has six digits. In base 2, the log tells you how many times you can halve a number before it reaches 1, which is why binary search through a million items takes only about 20 steps.

Where logarithms appear

Logs turn exponential growth into straight lines, which is why charts of populations, prices and virus cases often use a log scale. In finance, the time to double money at rate r is ln(2) ÷ ln(1 + r), which gives the rule of 72. In chemistry, pH is the negative base-10 log of hydrogen ion concentration. Sound level in decibels is 10 times the base-10 log of a power ratio. Each of these is a quick job for the log calculator.

Related converters and calculators

Related calculators:

• Exponent calculator: powers
• Scientific calculator: functions
• Scientific notation calculator: powers of 10

Sources

NIST Digital Library of Mathematical Functions, §4.2: Logarithms.

Formula and method

log_b(x) = ln(x) ÷ ln(b); base 10 and 2 use exact log10/log2; result rounded to 12 significant digits and snapped to integers.

Change-of-base formula without floating-point artifacts such as 2.9999999999999996. x > 0; b > 0 and b ≠ 1.

Questions about this calculator