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Logarithm calculator →

Logarithm guide

The Logarithm Calculator computes base-10, base-e (natural), and custom-base logarithms. It also demonstrates the laws of logarithms and visualizes logarithmic scales like decibels and pH.

Worked examples

Calculate a basic logarithm

  1. Open the "Basic Log" tab.
  2. Enter 10 as the Base.
  3. Enter 1000 as the Number (x).
Result: log₁₀(1000) = 3

A logarithm asks "To what power must the base be raised to get this number?" Since 10³ = 1000, the answer is 3.

Demonstrate the product rule

  1. Open the "Log Laws" tab.
  2. Enter 100 for Number A.
  3. Enter 10 for Number B.
Result: log(100 × 10) = log(100) + log(10) = 3

The product rule states that the log of a product equals the sum of the logs. Since log(1000) = 3, and log(100)+log(10) = 2 + 1 = 3.

Calculate decibels (logarithmic scale)

  1. Open the "Log Scales" tab.
  2. Select "decibel".
  3. Enter 60.
Result: Power ratio = 1,000,000×

Decibels are logarithmic. A 60 dB increase means the power is 10^(60/10) = 10^6 times stronger.

Tips

  • •The base must be positive and cannot equal 1.
  • •You cannot take the logarithm of zero or a negative number in real math.
  • •A base of 10 is the "common log" (often written as just log), and a base of e is the "natural log" (written as ln).

Frequently asked questions

What is a logarithm?
A logarithm is the inverse operation to exponentiation. It answers the question: "To what power must the base be raised to produce a given number?"
What is a natural logarithm (ln)?
A logarithm with the base e (Euler’s number, approximately 2.71828). It is heavily used in calculus because its derivative is simply 1/x.
What is a common logarithm (log)?
A logarithm with base 10. It is used in scientific notation, the pH scale, and the Richter scale.
Why can’t I take the log of a negative number?
Because no positive base raised to any real power will ever yield a negative result. Logs of negative numbers involve complex (imaginary) numbers.
What is log(0)?
It is undefined. As the number approaches 0, the logarithm approaches negative infinity.
What is log(1)?
It is always 0, for any valid base, because any number raised to the power of 0 is 1.
What is the product rule for logarithms?
log(x × y) = log(x) + log(y). Multiplication inside the log becomes addition outside.
What is the quotient rule for logarithms?
log(x / y) = log(x) - log(y). Division inside the log becomes subtraction outside.
What is the power rule for logarithms?
log(x^n) = n × log(x). An exponent inside the log can be moved to the front as a multiplier.
How do I change the base of a logarithm?
Use the formula log_b(x) = log(x) / log(b). The calculator handles this automatically when you enter a custom base.
What is a logarithmic scale?
A way of displaying data where the values span many orders of magnitude. Instead of equal steps, each step represents a multiplication (usually by 10).
What is a decibel (dB)?
A logarithmic unit used to measure sound intensity. Adding 10 dB means the sound power increases by a factor of 10.
What is the Richter scale?
A logarithmic scale used to measure earthquake magnitude. A magnitude 6 earthquake releases about 31.6 times more energy than a magnitude 5.
What is pH?
A logarithmic scale measuring acidity. pH = -log[H⁺]. A lower pH means a higher concentration of hydrogen ions (more acidic).
Why do graphs of logarithms curve downward?
Because as x gets very large, the exponent needed to reach it grows very slowly. This is why a log curve flattens out, growing to infinity at a decreasing rate.
What is the derivative of ln(x)?
The derivative is 1/x. This is why the natural log is so important in calculus.
What is the domain of a logarithm?
The domain is strictly positive real numbers (x > 0).
Can the base of a logarithm be a fraction?
Yes, as long as it is greater than 0 and not equal to 1. A base between 0 and 1 will produce a decreasing function.
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