Log Calculator

In mathematics, the logarithm (commonly abbreviated as log) is the inverse operation of exponentiation. If you know that a base (b) raised to the power of y equals x, the logarithm tells you what power you must apply:

log_b(x) = y ⇒ b^y = x

Where b is the base, x is the argument, and y is the exponent. The logarithm answers the question: “To what power must I raise the base in order to get this number?”

Our free Log Calculator solves for any of the three variables in the logarithmic equation log_b(x) = y. It supports standard decimal bases, the natural base e (natural log, ln), and binary base-2 calculations.

Common Logarithm Bases & Applications

While the base of a logarithm can technically be any positive real number except 1, three bases are conventionally used across scientific disciplines:

• Base 10 (Common Logarithm): Typically written as just log(x) with the base 10 implied. Used widely in science and engineering to measure magnitude scales like earthquake intensity (Richter scale), sound levels (decibels), and chemical acidity (pH).
• Base e (Natural Logarithm): Written as ln(x) where the base is Euler’s number e ≈ 2.7182818. Essential in physics, calculus, economics (for compound interest), and models of population growth.
• Base 2 (Binary Logarithm): Written as log_2(x). Crucial in computer science, information theory, and digital data storage algorithms.

Fundamental Logarithm Rules & Laws

To manipulate and simplify logarithmic expressions manually, mathematicians rely on these core algebraic laws:

1. Product Rule

The logarithm of a product is the sum of the logarithms of its factors:

log_b(x × y) = log_b(x) + log_b(y)

Example: log(1 × 10) = log(1) + log(10) = 0 + 1 = 1.

2. Quotient Rule

The logarithm of a quotient is the difference of the logarithms of the numerator and denominator:

log_b(x / y) = log_b(x) - log_b(y)

Example: log(10 / 2) = log(10) - log(2) = 1 - 0.301 = 0.699.

3. Power Rule

An exponent in the argument of a logarithm can be moved to the front as a multiplier:

log_b(x^y) = y × log_b(x)

Example: log(2^6) = 6 × log(2) ≈ 6 × 0.301 = 1.806.

4. Change of Base Rule

Convert a logarithm from one base (b) to another base (k) using this ratio:

log_b(x) = log_k(x) / log_k(b)

Example: Convert log_10(x) using base 2:

log_10(x) = log_2(x) / log_2(10).

5. Base-Argument Swap Rule

Reciprocating a logarithm swaps the base and the argument:

log_b(c) = 1 / log_c(b)

Example: log_5(2) = 1 / log_2(5).

Key Logarithmic Identities & Asymptotes

Keep these special values and boundaries in mind when calculating logarithms:

• log_b(1) = 0 (Since b^0 = 1 for any non-zero base).
• log_b(b) = 1 (Since b^1 = b).
• log_b(0) = Undefined (You cannot raise a positive base to any power to get exactly 0).
• ln(e^x) = x (The natural logarithm and exponential function cancel out).
• Asymptote Limit: As the argument x approaches 0 from the positive side, the logarithm approaches negative infinity:
  
  lim_(x → 0^+) log_b(x) = -∞ (for base b > 1).

Solve exponential growth equations using our Exponent Calculator or handle advanced operations with the Scientific Calculator.

Frequently Asked Questions (FAQ)

What is the difference between log and ln?

log conventionally refers to the common logarithm, which uses a base of 10. ln stands for natural logarithm (logarithmus naturalis), which uses a base of Euler’s number e (approximately 2.71828).

Can you take the logarithm of a negative number?

In the set of real numbers, the logarithm of a negative number is undefined. This is because a positive base raised to any real power will always yield a positive result. In advanced mathematics, logarithms of negative numbers can be calculated using complex numbers.

Why is log of 0 undefined?

A logarithm asks: “To what power do I raise base b to get argument x?” If x = 0, there is no real exponent y such that b^y = 0 (since a positive base multiplied by itself will never equal zero). Therefore, log_b(0) has no real solution.

What is the natural base e?

Euler’s number e is an irrational constant approximately equal to 2.718281828459. It is the base of natural logarithms and represents the limit of (1 + 1/n)^n as n approaches infinity, which is the foundational formula for continuous growth and compounding interest.