Log Calculator

A logarithm answers the question: "what power do I raise the base to in order to get this number?" If 10^2 = 100, then log₁₀(100) = 2. The concept extends to any positive base and any positive input. Three bases dominate practical use. Base 10 (common log) is used for pH, decibels, and earthquake magnitudes. Base e (natural log) appears throughout calculus, statistics, and physics. Base 2 (binary log) is fundamental to information theory and computing, where it counts the number of bits needed to represent a value. This calculator shows all four simultaneously. The underlying math uses the change-of-base formula: log_b(x) = ln(x) / ln(b), so every result is derived from the natural log.

• pH and chemistry: pH = -log₁₀[H⁺] where [H⁺] is the hydrogen ion concentration. Enter the concentration to find pH, or check how pH relates to concentration.
• Decibels in audio and acoustics: Sound intensity in decibels: dB = 10 × log₁₀(I/I₀). Enter the intensity ratio to compute decibels.
• Binary information content: The number of bits needed to represent n equally likely outcomes is log₂(n). For 256 colors: log₂(256) = 8 bits, confirming 8-bit color.
• Exponential growth and decay: Solving for time in exponential models like A = A₀ × e^(kt) requires the natural log. Enter A/A₀ and compute ln to find the exponent.

1. 1Enter the positive number you want to find the logarithm of.
2. 2Select the primary base: 10, e, 2, or custom.
3. 3For a custom base, enter the base value in the custom base field.
4. 4The calculator computes the natural log of x first, then divides by the natural log of each base.
5. 5All four logarithm values are displayed regardless of the selected primary base.

• Logarithm of zero or negative numbers: log(0) is negative infinity, and the log of a negative number is undefined in real numbers. The calculator only accepts positive inputs. If you have a negative value, consider whether you should use its absolute value.
• Confusing ln and log: In engineering and science, "log" often means base 10. In mathematics and many textbooks, "log" means natural log (base e). This calculator shows both explicitly to avoid ambiguity.
• Using base 1 for a custom log: The base 1 is invalid for logarithms: 1^y = 1 for all y, so there is no unique y that satisfies 1^y = x for x ≠ 1. The calculator returns an error for base 1.
• Forgetting the change-of-base formula: When computing log_5(125) by hand, use log_5(125) = ln(125)/ln(5) = 4.828/1.609 = 3. Enter 125 with custom base 5 in this calculator to verify.

• Use base 10 for order-of-magnitude estimates: The integer part of log₁₀(x) tells you how many digits the number has. log₁₀(1,500,000) ≈ 6.18, so it is a 7-digit number. Quick sanity checks for large numbers.
• Use ln for continuous growth problems: When a problem involves e^x (compound interest with continuous compounding, radioactive decay, etc.), the inverse uses ln. ln(A/A₀) = kt gives you time directly.
• Binary log for information theory: log₂(n) gives the number of bits needed to distinguish n equally likely outcomes. This is Shannon entropy for uniform distributions and the basis for compression analysis.
• Verify: log_b(b) always equals 1: A quick sanity check: the log of any base to itself is always 1, because b^1 = b. Enter any base and set the value equal to the base. The result should be exactly 1.