📈Logarithm Calculator
Solve any logarithm equation log_b(x) = y by entering any two of the three values. Supports any base including e and 10. Shows natural log, log₁₀, log₂, antilog, and step-by-step explanation.
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Enter any two of the three values — the third will be solved automatically.
Leave one field blank. Accepts e as base for natural log, or any positive number.
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Logarithm Calculator: Solve log_b(x) = y for Any Unknown
This logarithm calculator solves the equation log_b(x) = y for any of the three unknowns. Enter any two values — the argument x, the base b, or the result y — and the calculator finds the third. It supports any positive base, including e (natural log) and 10 (common log), and shows step-by-step working using the change-of-base formula and exponential form.
What Is a Logarithm?
A logarithm answers the question: to what power must you raise the base to obtain a given number? If b^y = x, then log base b of x equals y. For example, log base 10 of 1,000 equals 3 because 10^3 = 1,000. Logarithms are the inverse operation of exponentiation in the same way that subtraction is the inverse of addition.
Solving for x, b, or y
The equation log_b(x) = y has three variables. Knowing any two determines the third:
- Solve for y (standard log): y = ln(x) / ln(b) using the change-of-base formula.
- Solve for x (antilogarithm): x = b^y. This reverses the logarithm — known as the antilog.
- Solve for b (find the base): b = x^(1/y). Useful when you know the input and output but need the base.
Natural Log Calculator: ln and Euler's Number
The natural logarithm, written ln(x), uses Euler's number e (approximately 2.71828) as its base. It arises naturally in calculus: the derivative of ln(x) is exactly 1/x, and e^x is the unique function whose rate of change equals itself. Enter "e" in the base field to use the natural log.
Log Base 10 and Log Base 2
The common logarithm (base 10) describes the order of magnitude of a number and powers applications like pH, decibels, and the Richter scale. The binary logarithm (base 2) measures information in bits and is foundational in computer science — log₂(n) is the number of binary digits needed to represent n states.
Change of Base Formula
To compute any logarithm, use: log_b(x) = ln(x) / ln(b). This converts any base into a ratio of natural logs. For example, log base 3 of 81 = ln(81)/ln(3) = 4.382/1.099 = 4. Verify: 3^4 = 81.
Antilog: Reversing a Logarithm
The antilogarithm reverses the logarithm. If log₁₀(x) = y, then x = 10^y. This calculator computes the antilog automatically when you enter b and y and leave x blank.
Frequently Asked Questions
What is the difference between log and ln?
log (also written log10) is the base-10 logarithm, commonly used in science and engineering. ln is the natural logarithm with base e (approximately 2.71828), used in calculus, statistics, and finance. They are related by ln(x) = log10(x) × ln(10) ≈ log10(x) × 2.3026.
How do I calculate a logarithm for a custom base?
Use the change-of-base formula: log_b(x) = ln(x) / ln(b). Enter your x and b values and leave y blank — the calculator applies this formula automatically. For example, log base 5 of 125 = ln(125)/ln(5) = 4.828/1.609 = 3. Verify: 5^3 = 125.
How do I solve for the base of a logarithm?
If log_b(x) = y, then b = x^(1/y). For example, if log_b(81) = 4, then b = 81^(1/4) = 3. Enter x and y and leave b blank to have this calculator solve it automatically.
What is the antilogarithm?
The antilogarithm reverses a logarithm. If log_b(x) = y, then the antilog is x = b^y. To compute it, enter b and y and leave x blank. For example, if log₁₀(x) = 3, then x = 10^3 = 1000.