How to Use This Calculator
Say you want to solve the equation 2ˣ = 64. Enter number = 64 and base = 2. The result is log₂(64) = 6, because 2⁶ = 64. The natural log ln(64) = 4.158883, log₁₀(64) = 1.80618, and log₂ is already shown. The antilog confirms 2⁶ = 64. Now try something more practical: the pH of a solution is -log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration. If [H⁺] = 3.2 × 10⁻⁵, enter number = 0.000032, base = 10, and the result is -4.49485, so the pH is about 4.49 — that's acidic, like orange juice. If you change the number to 1 × 10⁻⁷, log₁₀(10⁻⁷) = -7, meaning pH = 7, which is neutral water.
About Log Calculator
The Log Calculator computes the logarithm of a number in any base, plus the natural logarithm (ln), common logarithm (log₁₀), binary logarithm (log₂), and the antilog (base raised to the result). Logarithms are the inverse of exponentiation and are fundamental across science, engineering, and finance. Use this tool to solve exponential equations, measure earthquake magnitudes on the Richter scale, calculate pH in chemistry, analyze sound intensity in decibels, or model population growth and radioactive decay. The calculator accepts any positive base (except 1) and any positive number, returning results accurate to six decimal places. The change-of-base formula is applied internally so you can work with any base effortlessly. Seeing all four log forms side by side helps users recognize the relationship between different logarithmic scales. Whether you are a college student in calculus or chemistry, a data scientist working with log-transformed variables, or a hobbyist exploring mathematical functions, this tool provides quick and precise logarithmic calculations.
When to Use This Calculator
Chemistry (pH & pKa): The pH scale is logarithmic: each whole number change represents a tenfold change in acidity. A solution with pH 3 is 10 times more acidic than pH 4 and 100 times more than pH 5. Enter the hydrogen ion concentration to find pH instantly, or work backwards from pH to find concentration.
Earthquake & Sound Measurement: The Richter scale is logarithmic — a magnitude 6 earthquake is 10 times more powerful than magnitude 5. Similarly, decibels measure sound intensity logarithmically. A 70 dB sound is 10 times louder than 60 dB. Use the log calculator to compare values on these exponential scales.
Finance & Population Growth: If an investment doubles every 10 years, how long to grow by 8 times? That's log₂(8) = 3 doubling periods = 30 years. For continuous compounding, the natural log ln tells you the time needed. A population growing at 2% per year doubles when ln(2)/0.02 ≈ 34.7 years.