Log Calculator
log_b(x) = y where b^y = x
Example:
log₂(8) = 3 since 2^3 = 8
What Is a Logarithm?
A logarithm (or log) is the inverse function of exponentiation. In simple terms, it helps you find the power you need to raise a specific base to in order to get a certain number.
For example:
If x = bʸ, then y = log_b(x)
Here, b is the base, x is the result, and y is the exponent (or logarithm).
By default, log(x) usually refers to base 10 (common logarithm), while ln(x) represents the natural logarithm with base e. You’ll also come across log₂(x) in computer science, where the base is 2.
Common Bases Used:
Base 10: Popular in science and engineering.
Base e (~2.718): Common in mathematics and physics.
Base 2: Widely used in computing and binary systems.
Basic Logarithm Rules
Understanding logarithmic rules can make working with logs easier. Here are the key properties:
Product Rule
The logarithm of a product is the sum of the individual logs:
log_b(x × y) = log_b(x) + log_b(y)
Example:
log(1 × 10) = log(1) + log(10) = 0 + 1 = 1
Quotient Rule
The log of a division is the difference of the logs:
log_b(x / y) = log_b(x) – log_b(y)
Example:
log(10 / 2) = log(10) – log(2) ≈ 1 – 0.301 = 0.699
Power Rule
An exponent inside a log can be moved to the front:
log_b(xʸ) = y × log_b(x)
Example:
log(2⁶) = 6 × log(2) ≈ 1.806
Logarithmic Base Conversion
You can convert a logarithm from one base to another using this formula:
log_b(x) = log_k(x) / log_k(b)
Example:
log₁₀(x) = log₂(x) / log₂(10)
Reciprocal Rule (Switching Base and Argument)
You can also reverse the base and the argument:
log_b(c) = 1 / log_c(b)
Example:
log₅(2) = 1 / log₂(5)
Special Logarithm Values to Remember
log_b(1) = 0
log_b(b) = 1
log_b(0) = undefined
As x → 0, log_b(x) → -∞
ln(eˣ) = x