Logarithm Calculator

Easily calculate the logarithm of a number to a specified base.

Enter the number and the base to find the logarithm. The logarithm y = log_b(x) is the power to which the base b must be raised to get the number x.

Practical Examples

Use these examples to see how the calculator works with different inputs.

Common Logarithm (Base 10)

common-log

Calculating the common logarithm of 1000.

Number (x): 1000

Base (b): 10

Binary Logarithm (Base 2)

binary-log

Calculating the binary logarithm of 16.

Number (x): 16

Base (b): 2

Natural Logarithm (Base e)

natural-log

Calculating the natural logarithm of e² (approx 7.389).

Number (x): 7.389056

Base (b): 2.71828

Custom Base Logarithm

custom-base

Calculating the logarithm of 625 with base 5.

Number (x): 625

Base (b): 5

Other Titles
Understanding the Logarithm Calculator: A Comprehensive Guide
Explore the fundamentals of logarithms, their properties, and how to use this calculator effectively for your mathematical needs.

What is a Logarithm?

  • Core Definition
  • Common and Natural Logs
  • The Logarithmic Identity
A logarithm is the mathematical operation that is the inverse of exponentiation. It answers the question: 'To what exponent must we raise a given base to obtain a certain number?'
The relationship is formally stated as: y = log_b(x) is equivalent to b^y = x. Here, 'b' is the base, 'x' is the argument, and 'y' is the logarithm.
Key Logarithm Types
1. Common Logarithm: A logarithm with base 10 (log₁₀). It's so common that if you see 'log(x)' without a specified base, the base is assumed to be 10.
2. Natural Logarithm: A logarithm with base 'e' (Euler's number, ≈ 2.71828), written as 'ln(x)'. It's fundamental in calculus and sciences modeling growth and decay.

Fundamental Examples

  • log₁₀(100) = 2, because 10² = 100.
  • log₂(8) = 3, because 2³ = 8.
  • ln(e) = 1, because e¹ = e.

Step-by-Step Guide to Using the Logarithm Calculator

  • Inputting Your Values
  • Interpreting the Result
  • Using the Reset Function
1. Enter the Number (x)
In the 'Number (x)' field, type the positive number you want to find the logarithm of.
2. Enter the Base (b)
In the 'Base (b)' field, enter the base of the logarithm. Remember, the base must be a positive number and cannot be 1.
3. Calculate and View the Result
Click the 'Calculate' button. The calculator will display the logarithm in the 'Result' section. If your inputs are invalid (e.g., a negative number), an error message will appear to guide you.

Practical Walkthroughs

  • To find log₃(81): Enter Number (x) = 81, Base (b) = 3. The result will be 4.
  • To find ln(1): Enter Number (x) = 1, Base (b) = 2.71828. The result will be 0.

Real-World Applications of Logarithms

  • Science and Engineering
  • Finance and Economics
  • Computer Science
Measuring Intensity: pH, Decibels, and Richter Scale
Logarithmic scales are used to manage and represent huge ranges of values. The pH scale (acidity), decibel scale (sound intensity), and Richter scale (earthquake magnitude) are prime examples. A small step on these scales represents a massive leap in real-world quantity.
Financial Growth and Compound Interest
Logarithms help calculate the time it takes for an investment to grow to a certain amount under compound interest, a cornerstone of financial planning.
Algorithmic Complexity in Computer Science
In computer science, the efficiency of many algorithms is described using logarithms (e.g., O(log n)). This signifies that the time taken to run the algorithm increases slowly as the input size grows, which is highly desirable.

Logarithms in Action

  • A pH of 3 is 10 times more acidic than a pH of 4.
  • A 6.0 earthquake is 100 times more powerful than a 4.0 earthquake.
  • A binary search algorithm has a time complexity of O(log n), making it very efficient for searching large datasets.

Common Misconceptions and Correct Methods

  • Addition and Subtraction Rules
  • Power and Root Rules
  • Change of Base
Misconception: log(x + y) = log(x) + log(y)
This is incorrect. The correct property is the product rule: logb(xy) = logb(x) + log_b(y). There is no simple formula for the logarithm of a sum.
Misconception: log(x) / log(y) = log(x - y)
This is also incorrect. The correct property is the quotient rule: logb(x/y) = logb(x) - log_b(y).
Misconception: (log(x))^n = n*log(x)
This confuses the power of a log with the log of a power. The correct power rule is logb(x^n) = n * logb(x).

Avoiding Common Pitfalls

  • Correct: log₂(4*8) = log₂(4) + log₂(8) = 2 + 3 = 5.
  • Correct: log₁₀(100/10) = log₁₀(100) - log₁₀(10) = 2 - 1 = 1.
  • Correct: log₃(9²) = 2 * log₃(9) = 2 * 2 = 4.

Mathematical Derivation and Formulas

  • The Change of Base Formula
  • Derivation Steps
  • Practical Application
The Change of Base Formula
Most calculators only have buttons for common log (base 10) and natural log (base e). To find a logarithm with a different base, you must use the change of base formula.
The formula is: logb(x) = logc(x) / log_c(b). Here, 'c' can be any new base, but 10 or 'e' are the most convenient choices.
How It's Derived
  1. Start with y = log_b(x), which means b^y = x.
  2. Take the log base 'c' of both sides: logc(b^y) = logc(x).
  3. Apply the power rule: y * logc(b) = logc(x).
  4. Solve for y: y = logc(x) / logc(b).

Formula in Practice

  • To calculate log₂(7) using a calculator with 'ln': log₂(7) = ln(7) / ln(2) ≈ 1.9459 / 0.6931 ≈ 2.807.
  • To calculate log₅(100) using a calculator with 'log': log₅(100) = log(100) / log(5) = 2 / 0.69897 ≈ 2.861.