Logarithm Calculator
Must be a positive number greater than zero
Enter Your Values
Choose a mode above, enter your values, and your logarithm results will appear here instantly with step-by-step workings and a visual chart.
Logarithm Properties Reference
logb(x·y) = logb(x) + logb(y)Log of a product equals sum of logslogb(x/y) = logb(x) − logb(y)Log of a quotient equals difference of logslogb(xⁿ) = n · logb(x)Log of a power equals exponent times loglogb(x) = ln(x) / ln(b)Convert to any computable baselogb(b) = 1Log of the base itself equals 1logb(1) = 0Log of 1 always equals 0b^(logb(x)) = xExponentiation and log cancel each other outHow to Use the Logarithm Calculator
Choose Your Mode
Select one of four modes at the top: Standard Log for computing log_b(x) directly, Equation Solver for solving log equations step-by-step, Log Table to generate a range of log values for export, or Applications for real-world calculators like pH, decibels, Richter scale, doubling time, and investment growth.
Enter Your Inputs
In Standard mode, enter your number and pick a base (2, e, 10, or custom). In Equation Solver, choose the equation pattern (e.g. sum, coefficient), enter the base, the right-hand side constant, and any secondary value. In Log Table mode, specify the base, start, end, and step size. In Applications, select the sub-calculator and fill in the appropriate fields.
Review the Results and Steps
Results appear automatically as you type. Standard mode shows the log value, antilog, equivalent exponential form (b^y = x), and a multi-base comparison bar chart. The Equation Solver and Applications modes display numbered step-by-step workings so you can follow the mathematical logic. The log curve chart visualizes the full log function with your input point highlighted.
Export or Copy
Click 'Copy Result' to copy the computed value to your clipboard for use in other applications. In Log Table mode, click 'Export CSV' to download the full table as a comma-separated file that opens in Excel, Google Sheets, or any spreadsheet application.
Frequently Asked Questions
What is the difference between log, ln, and log₂?
All three are logarithms, but with different bases. 'log' (without a base specified) conventionally means base 10 — the common logarithm used in science, chemistry, and engineering. 'ln' is the natural logarithm, base e ≈ 2.71828 — the base that appears naturally in calculus, growth models, and physics. 'log₂' is the binary logarithm, base 2 — the natural unit in computer science, information theory, and digital signal processing. All three can be converted to each other using the change-of-base formula: log_b(x) = ln(x) / ln(b). This calculator computes all three simultaneously in Standard mode so you can compare them for any input.
Why is the logarithm of zero or a negative number undefined?
Logarithms are defined only for positive arguments because they are the inverse of exponentiation with a positive base. Since b^y is always positive for any finite exponent y (when b > 0 and b ≠ 1), there is no real exponent y that satisfies b^y = 0 or b^y = negative number. As x approaches zero from the positive side, log(x) approaches negative infinity, but it never reaches an actual value at zero. For negative numbers, logarithms require complex numbers (the imaginary unit appears), which are outside the scope of real-number computation this calculator performs. The tool will display 'Undefined' for such inputs.
How does the equation solver handle different patterns?
The equation solver applies standard logarithm rules algebraically. For log_b(x) = c it uses the definition directly to get x = b^c. For the sum pattern (log_b(x) + log_b(a) = c) it applies the Product Rule (the two logs combine into one log of a product), then exponentiates to get x·a = b^c, and finally divides. For the coefficient pattern (k·log_b(x) = c) it applies the Power Rule to turn the coefficient into an exponent, yielding x = b^(c/k). Each step is displayed in the numbered step-by-step breakdown so you can follow the algebra and verify the result.
How is pH calculated using logarithms?
pH measures the acidity or alkalinity of a solution using the formula pH = −log₁₀([H⁺]), where [H⁺] is the hydrogen ion concentration in moles per litre. Neutral water at 25°C has [H⁺] = 10⁻⁷ mol/L, giving pH = −log₁₀(10⁻⁷) = 7. Acids have higher H⁺ concentrations and lower pH values (below 7), while bases have lower H⁺ concentrations and higher pH values (above 7). The logarithmic scale means each unit change in pH corresponds to a 10× change in hydrogen ion concentration. Enter the H⁺ concentration in the Applications mode pH sub-calculator — scientific notation (e.g. 1e-7) is supported.
What is doubling time and how is it calculated?
Doubling time is the period required for a quantity growing at a constant percentage rate to double in size. Under continuous compounding, the formula is t = ln(2) / r, where r is the annual growth rate expressed as a decimal (e.g. 7% = 0.07). The natural logarithm of 2 (≈ 0.6931) reflects the factor of doubling. For example, at 7% annual growth, the doubling time is ln(2) / 0.07 ≈ 9.9 years. A well-known approximation is the Rule of 72: divide 72 by the percentage rate to estimate doubling time. The calculator's Applications mode computes exact values and shows the full derivation.
How do I generate and export a logarithm table?
Switch to Log Table mode. Enter your desired base, a start value (must be positive), an end value, and a step size. The calculator generates a table of x and log_b(x) pairs for each step, up to a maximum of 200 rows. The table appears instantly in the results panel. Click 'Export CSV' to download the table as a .csv file named 'log-table-base-N.csv'. This file can be opened directly in Microsoft Excel, Google Sheets, LibreOffice Calc, or any text editor. The CSV format has two columns: x and log_b(x) with the computed value to six decimal places.