Calculate log base 10, natural log (ln), log base 2, and any custom base logarithm.
A logarithm is the inverse operation of exponentiation. If bⁿ = x, then logᵦ(x) = n. In other words, the logarithm answers the question: 'To what exponent must b be raised to produce x?' Logarithms are central to mathematics, physics, information theory, and signal processing.
The Logarithm Calculator by Calculator Expert computes the common logarithm (log₁₀), natural logarithm (ln, base e), binary logarithm (log₂), and logarithms in any custom base. It shows the antilog and the change-of-base calculation steps. Instantly useful for students, engineers, and scientists.
| Expression | Value | Explanation |
|---|---|---|
| log₁₀(1) | 0 | 10⁰ = 1 |
| log₁₀(10) | 1 | 10¹ = 10 |
| log₁₀(100) | 2 | 10² = 100 |
| ln(e) | 1 | e¹ = e |
| log₂(8) | 3 | 2³ = 8 |
Enter your values in the empty input fields above and click "Calculate." All fields start empty so you can input any values you need. The result is displayed instantly with the working formula. Calculator Expert provides accurate, ad-free calculations for students, teachers, and professionals.
Method 1: Change-of-Base Formula: Any logarithm can be computed using logᵦ(x) = log(x)/log(b) = ln(x)/ln(b). This is the standard method used by scientific calculators and computers.
Method 2: Inverse Exponential Method: Since logarithm and exponentiation are inverses, if bⁿ = x, then logᵦ(x) = n. Solve by finding what power of b gives x, using trial and successive approximation.
Logarithms are only defined for positive real numbers. log(0) and log(negative numbers) are undefined. The base must be positive and not equal to 1. Results for very small inputs close to 0 may be very large negative numbers. Irrational results are rounded to 6 decimal places.
In health metrics, the logarithmic body weight scale is sometimes used to normalize large differences in body mass across populations. The log transformation of weight data helps achieve a more normal distribution when computing BMI or Ponderal Index statistics across age groups.
Logarithms are used in decibel scale for sound, Richter scale for earthquakes, pH scale in chemistry, information entropy in data science, musical scales (frequency ratios), compound interest time calculations, and complexity analysis in computer algorithms (O(log n)).