Compute the antilogarithm: given a logarithm value and base, retrieve the original number. Supports common (base 10), natural (base e), or any custom base. Visualize the exponential curve y = bˣ and see the exact point (x, antilog).
The antilogarithm (antilog) is the inverse operation of a logarithm. If logb(y) = x, then the antilog of x with base b is y = bx. In simpler terms, the antilog raises the base to the power of the given logarithm value. This operation is fundamental in reversing logarithmic transformations, widely used in science, finance, and data normalization.
Antilogb(x) = bx
where b > 0, b ≠ 1, and x ∈ ℝ
The restrictions b > 0 and b ≠ 1 are fundamental to the definition of the logarithm and its inverse. The logarithm function y = logb(x) is defined only for a positive base b > 0, b ≠ 1 and a positive argument x > 0. Since the antilogarithm bx is the inverse function, it accepts any real exponent x, but the base b must maintain the same restrictions to ensure the output is a unique positive real number, preserving the one-to-one correspondence between the logarithmic and exponential forms.
Given a base b (positive, not 1) and a logarithmic value x, the antilog is simply exponentiation: bx. For natural antilog (base e), we compute exp(x). For common antilog (base 10), compute 10x. For any custom base, use pow(b, x). The tool handles floating-point precision using IEEE 754 double arithmetic, accurate to about 15 decimal digits.
The exponential function grows (or decays) rapidly. Our interactive graph plots y = bx across a dynamically adjusted domain centered on x, helping you visualize the relationship between the logarithm input and the antilog output.
Antilogarithms for any positive base b can be expressed using the natural exponential function. Using the change-of-base property: bx = ex·ln(b). This identity underpins the universal computation of exponential functions and is used internally in many mathematical libraries. It connects the natural antilog (exp) to antilogs of any base.
| Base b | Log value x | Antilog (bˣ) | Application context |
|---|---|---|---|
| 10 | 3 | 1000 | Orders of magnitude: 10³ = thousand |
| e (2.71828) | 2.302585 | 10.0 | Converting ln(10) back to 10 |
| 2 | 10 | 1024 | Computer memory (2¹⁰ bytes = 1 KB) |
| 10 | -2 | 0.01 | pH calculation: [H⁺] = 10⁻² = 0.01 M |
| 3 | 4 | 81 | Exponential growth: 3⁴ = 81 |
In pharmacokinetics, drug concentration often decays logarithmically. A model gives log₁₀(C) = 1.5 after 2 hours. To find concentration C, we compute antilog₁₀(1.5) = 101.5 ≈ 31.62 mg/L. This value is essential for determining therapeutic windows. Our calculator instantly provides the precise concentration, aiding clinical decisions and research validation.
In finance, the formula for continuous compound interest is A = P·ert, where P is the principal, r is the annual interest rate, and t is time in years. Suppose we have an investment of $1000 at an annual rate of 5% for 3 years. The continuous compounding factor is e0.05×3 = e0.15. Using the natural antilog, we find e0.15 ≈ 1.16183. Therefore, the future value is $1000 × 1.16183 = $1161.83. This example shows how the antilog (exponential) function is used to calculate the growth of an investment.
Logarithms were introduced by John Napier in 1614 as a way to simplify calculations, especially in astronomy and navigation. The antilogarithm, as the inverse operation, was equally important. Before the advent of calculators, logarithms and antilogarithms were computed using logarithm tables. The concept of the natural logarithm and the base e was developed later by Leonhard Euler, who discovered the constant e ≈ 2.718281828459045. The antilogarithm function, especially the natural antilogarithm exp(x), is central to calculus and differential equations.
The natural antilog function exp(x) = eˣ is intimately linked to Euler's number e ≈ 2.718281828459045. The exponential function is its own derivative, making it central to calculus, differential equations, and probability (normal distribution). Our calculator uses the built-in Math.exp and Math.pow for high-precision evaluation.