Log Calculator (Logarithm)
Please provide any two values to calculate the third in the logarithm equation logbx = y. It can accept "e" as a base input.
What is Log?
A logarithm is the inverse operation of exponentiation. In other words, if we have an exponential equation x = by, then the logarithmic form is y = logb x, where b is the base.
Logarithms answer the question: "To what power must we raise the base to get the argument?" For example, log2(8) = 3 because 2³ = 8.
Common Bases:
- Base 10 (Common Logarithm): log10(x) or simply log(x) - used in many scientific and engineering applications
- Base e (Natural Logarithm): loge(x) or ln(x) - where e ≈ 2.71828, fundamental in calculus and natural sciences
- Base 2 (Binary Logarithm): log2(x) - commonly used in computer science and information theory
General Form:
x = by; then y = logb x; where b is the base
Basic Log Rules
These fundamental rules of logarithms are essential for simplifying logarithmic expressions and solving logarithmic equations:
Product Rule
logb(x × y) = logb x + logb y
EX: log(1 × 10) = log(1) + log(10) = 0 + 1 = 1
The logarithm of a product is the sum of the logarithms of the factors.
Quotient Rule
logb(x / y) = logb x - logb y
EX: log(10 / 2) = log(10) - log(2) = 1 - 0.301 = 0.699
The logarithm of a quotient is the difference of the logarithms of the numerator and denominator.
Power Rule
EX: log(2⁶) = 6 × log(2) = 1.806
The logarithm of a power is the exponent times the logarithm of the base.
Change of Base Rule
logb(x) = logk(x) / logk(b)
EX: log₁₀(x) = log₂(x) / log₂(10)
This rule allows you to convert a logarithm from one base to another. The base k can be any positive number (commonly 10 or e).
Reciprocal Rule (Switch Base and Argument)
EX: log₅(2) = 1 / log₂(5)
This rule shows the reciprocal relationship when you switch the base and argument of a logarithm.
Other common logarithms to take note of include:
logb(1) = 0
The logarithm of 1 is always 0, regardless of the base, because any number raised to the power of 0 equals 1.
logb(b) = 1
The logarithm of the base equals 1, because any number raised to the power of 1 equals itself.
logb(0) = undefined
The logarithm of 0 is undefined for any base, because there is no power to which you can raise a positive number to get 0.
limx→0 logb(x) = -∞
As x approaches 0 from the positive side, the logarithm approaches negative infinity.
ln(ex) = x
The natural logarithm of e raised to any power x equals x. This is a fundamental property of the natural logarithm function.
Applications of Logarithms
Science and Engineering
- pH scale in chemistry (pH = -log[H⁺])
- Richter scale for earthquake magnitude
- Decibel scale for sound intensity
- Radioactive decay calculations
Computer Science
- Binary search algorithms (log₂ complexity)
- Information theory and entropy
- Data compression algorithms
- Complexity analysis of algorithms
Finance
- Compound interest calculations
- Investment growth modeling
- Time value of money
- Exponential growth and decay