About

This utility provides high-precision calculation of logarithmic and exponential functions, designed for engineers, students, and scientists. Unlike standard calculators, it contextualizes results with an interactive graph and step-by-step derivation using the Change of Base formula. It addresses the critical need for accuracy in fields like acoustics (decibels), information theory (entropy), and chemistry (pH), where logarithmic scales turn exponential magnitudes into manageable linear data.

The tool supports arbitrary bases, including standard constants like Euler's number (e) and binary (base 2). It features an integrated database of scientific presets, allowing users to quickly switch between contexts - calculating the decay of a radioactive isotope one moment and the bit-depth required for a data structure the next. Robust error handling ensures that domain violations (such as taking the log of a negative number) are explained mathematically, not just flagged as errors.

Formulas

The core relationship connects the base b, the exponent y, and the argument x. The logarithm asks: "To what power must the base be raised to yield the argument?"

log_b(x) = y ⇔ b^y = x

When calculating non-standard bases, the Change of Base Formula is essential. It allows conversion to a base supported by standard algorithms (usually Natural Log):

log_b(x) = ln(x)ln(b)

Other critical properties for simplifying expressions include:

{

log(uv) = log(u) + log(v)log(u/v) = log(u) − log(v)log(uⁿ) = n⋅log(u)

Reference Data

Domain	Formula / Context	Base (b)	Common Value
Mathematics	ln(x) (Natural Log)	e (2.718...)	Growth/Decay
Mathematics	log(x) (Common Log)	10	Orders of Magnitude
Computer Science	lb(x) (Binary Log)	2	Bits, Tree Height
Acoustics	Decibels (dB)	10^0.1 (1.258...)	Sound Pressure
Music Theory	Octaves	2	Frequency Doubling
Music Theory	Semitones	2^1/12 (1.059...)	Equal Temperament
Chemistry	pH Level	10	−log[H+]
Info Theory	Shannon Entropy	2	Bits of Info
Seismology	Richter Scale	10	Earthquake Amp.
Astronomy	Apparent Magnitude	100^1/5 (2.512...)	Star Brightness
Finance	Doubling Time	e	Compound Interest
Photography	Exposure Stops	2	Light Intensity

Frequently Asked Questions

In the real number system, a positive base raised to any power (positive, negative, or zero) always results in a positive number. Therefore, there is no real exponent "y" that solves b^y = x when x is negative. Complex logarithms (involving imaginary numbers) are required to solve such equations.

The logarithm of zero is undefined. However, as the input approaches zero from the positive side, the result approaches negative infinity (assuming the base is greater than 1). This is visually represented as a vertical asymptote at x = 0.

While calculators often default to base 10 or base e, you can calculate log base 2 using the Change of Base formula: ln(x) divided by ln(2). This tool features a dedicated "Computer Science (Base 2)" preset for this exact purpose.

A negative logarithmic result indicates that the input value (x) is between 0 and 1 (assuming the base is greater than 1). For example, log10(0.1) = -1. This corresponds to a fractional value: 10^-1 equals 1/10.

A base of 1 is a singularity. Since 1 raised to any power is always 1, the function f(y) = 1^y is a horizontal line, not a one-to-one function. It cannot be inverted to form a logarithm.