Calculate the logarithm of a value to any base, using the change-of-base formula, including common logarithm (base 10) and natural logarithm (base e ≈ 2.71828).
Google AdSense Banner
This area will contain advertisements after approval.
Google AdSense Banner
This area will contain advertisements after approval.
log_b(x) answers the question 'to what power must b be raised to get x?'. For example, log_10(100) = 2, since 10^2 = 100. This calculator uses the change-of-base formula, log_b(x) = ln(x) ÷ ln(b), which works for any valid base by expressing the result in terms of the natural logarithm.
The common logarithm uses base 10 (written log(x) without a base, often used in fields like chemistry for pH, and acoustics for decibels). The natural logarithm uses base e (approximately 2.71828, written ln(x)), which arises naturally in calculus, continuous compound growth, and many scientific formulas. To calculate a natural log with this calculator, enter 2.71828 as the base, or simply use base 10 for a common log.
log_10(100) asks 'to what power must 10 be raised to get 100?', the answer is 2, since 10² = 100. Similarly, log_2(8) asks the same question with base 2: 2³ = 8, so log_2(8) = 3. This calculator finds both using the change-of-base formula: log_2(8) = ln(8) ÷ ln(2) ≈ 2.0794 ÷ 0.6931 = 3.
Entering a negative number or zero as the value, logarithms of non-positive numbers are undefined in the real number system, this calculator requires a value greater than zero.
Forgetting what base to use for 'log' versus 'ln' in a textbook problem, plain 'log' conventionally means base 10 unless stated otherwise, while 'ln' always means the natural log (base e ≈ 2.71828), enter the matching base here to get the equivalent result.
Mixing up which number is the value and which is the base, log_2(8) and log_8(2) are reciprocals of each other (3 versus roughly 0.333), not the same number, so it matters which field each number goes into.
Since 1 raised to any power always equals 1, log base 1 is undefined, there's no consistent power that produces any value other than 1.
In the real number system, no real power of a positive base produces a negative result or zero, so logarithms of non-positive numbers are undefined (they do exist in the complex number system, but that's outside the scope of this calculator).
Logarithms and exponents are inverse operations, if b^y = x, then log_b(x) = y. Our Exponent Calculator handles the forward direction (calculating a power); this calculator handles the reverse.
Exponent Calculator
Calculate a base raised to any exponent, including negative and fractional exponents.
Scientific Notation Calculator
Convert between standard decimal notation and scientific notation.
Google AdSense Banner
This area will contain advertisements after approval.