Use this logarithm calculator to compute common logarithm (base 10), natural logarithm (base e), or logarithm with any custom base. Get step-by-step solutions instantly.
Select the type of logarithm, enter a positive number, and click Calculate to find the logarithm value.
A logarithm is the inverse operation to exponentiation. It answers the question: 'To what power must a given base be raised to produce a given number?' If b^y = x, then y is the logarithm of x to base b, written as y = log_b(x).
For example, since 10³ = 1000, the logarithm of 1000 to base 10 (the common logarithm) is 3, or log₁₀(1000) = 3. Similarly, since e² ≈ 7.389, the natural logarithm of 7.389 is approximately 2, or ln(7.389) ≈ 2.
Here are the key formulas and properties of logarithms:
The common logarithm (base 10) is often written without the base:
The natural logarithm (base e) is denoted by ln:
The change of base formula allows us to compute logarithms with any base:
Used in pH measurements, earthquake magnitude (Richter scale), sound intensity (decibels), and radioactive decay calculations.
Applied in signal processing, control systems, and information theory for measuring signal strength and data compression.
Used in compound interest calculations, modeling exponential growth, and analyzing returns on investments over time.
Essential in algorithm analysis (Big O notation), data compression, machine learning, and information theory.
For custom-base logs, choose any base greater than 0 except 1. If you are comparing powers and exponents, this tool works well alongside exponent and scientific calculators.
A base-10 logarithm asks what power of 10 produces the target value.
log10(1000) = 3
Because 10^3 = 1000, the logarithm equals 3.
Natural logs use base e and often appear in growth and decay formulas.
ln(e^2) = 2
The natural logarithm returns the exponent when the base is e.
When the base is not 10 or e, the calculator applies the change-of-base formula automatically.
log2(32) = 5
The result is 5 because 2^5 = 32.
In most calculator contexts, log means base 10 and ln means base e. Both describe exponents, but they use different bases depending on the problem.
A real logarithm is defined only for positive input values. Zero and negative numbers do not produce real-number logarithm results.
If the base were 1, every power of 1 would still equal 1, so it would be impossible to determine a unique logarithm result.
Use the change-of-base formula whenever you need a logarithm with a custom base and your calculator or software only supports log or ln directly.
