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A logarithm answers one question: what exponent turns a base into a given number? Enter your value and base into the calculator above, and the result appears instantly. This free log calculator handles base 10, base 2, base e (natural log), or any arbitrary base you choose.

A logarithm is the inverse of exponentiation. If b raised to the power y equals x, then log base b of x equals y. For example, because 10² = 100, log base 10 of 100 is 2.

This tool is a quick estimate and learning aid. For academic proofs or professional engineering sign-offs, verify results with your own methods.

How the Logarithm Calculator Works as a Step-by-Step Tool

The calculator takes two inputs: the argument (the number you want the logarithm of) and the base. It returns the exponent that connects them.

Here is what happens when you press calculate:

  1. You enter the argument of the logarithm (must be a positive number).
  2. You choose the base (common choices: 10, e, or 2).
  3. The tool applies the logarithm formula and displays the result.

Behind the scenes, it uses the change of base formula: log base b of x = ln(x) / ln(b). That single relationship lets the calculator handle any base, not just the standard ones.

Step-by-step solutions help you see the math, not just the answer. If you are learning algebra or preparing for an exam, trace each line to confirm you understand the process.

Calculate a Log with Common Logarithms (Base 10)

Common logarithms use base 10. When you see "log" written without a base in most textbooks and on scientific calculators, it means log base 10.

Is log10 and log the same? In most contexts, yes. Unless another base is specified, "log" defaults to base 10.

Quick reference values:

  • log(1) = 0, because 10⁰ = 1
  • log(10) = 1, because 10¹ = 10
  • log(100) = 2, because 10² = 100
  • log(1000) = 3, because 10³ = 1000

To calculate a log base 10 of a number that is not a clean power of 10, you need the formula or this calculator. For instance, log(50) ≈ 1.699. That means 10 raised to 1.699 is approximately 50.

Common logarithms appear frequently in science, finance, and engineering. The Richter scale, decibel measurements, and pH values all rely on base 10 logarithms.

Natural Logarithms: Calculate ln Step by Step

A natural logarithm uses base e, where e ≈ 2.71828. The notation is "ln" instead of "log."

Is log and ln the same? No. "log" typically means base 10. "ln" always means base e. They answer the same type of question (what exponent?) but with different bases.

What is ln on a calculator? It is the button that computes the natural logarithm. Press it, enter your number, and you get the exponent to which e must be raised.

Step-by-step example for ln(7.389):

  1. You are asking: e to what power equals 7.389?
  2. e² = 7.389 (approximately).
  3. So ln(7.389) ≈ 2.

Natural logarithms are essential in calculus, compound interest formulas, and exponential growth models. Any time a quantity grows or decays continuously, ln is the right logarithm to use.

Base 2 Log Calculation and Arbitrary Base Calculator

Base 2 logarithms are central to computer science. Every bit in a binary system represents a power of 2, so log base 2 tells you how many bits you need.

Common base 2 values:

  • log₂(1) = 0
  • log₂(2) = 1
  • log₂(8) = 3
  • log₂(1024) = 10

How to calculate a logarithm with an arbitrary base? Use this calculator. Enter any positive base (other than 1) and any positive argument. The tool applies the change of base formula automatically.

What are the 3 types of logarithms? The three most common types are:

  1. Common logarithm (base 10)
  2. Natural logarithm (base e)
  3. Binary logarithm (base 2)

Any other positive base is valid, though less common outside specialized fields.

Change of Base: How to Compute a Logarithm in Any Base

The change of base formula is the single most useful identity for logarithm calculation:

log base b of x = log(x) / log(b)

You can use common log (base 10) or natural log (ln) in the formula. Both give the same result.

This formula matters because most scientific calculators only have log (base 10) and ln (base e) buttons. To find the value of a logarithm in any other base, you convert.

Change of Base Computation for Base 2, Base 10, and Beyond

Example: Find log base 5 of 125.

  1. Rewrite using the change of base formula: log₅(125) = log(125) / log(5).
  2. log(125) ≈ 2.0969.
  3. log(5) ≈ 0.6990.
  4. Divide: 2.0969 / 0.6990 ≈ 3.
  5. Check: 5³ = 125. Correct.

Example: Convert a base 10 log to base 2.

  1. log₂(100) = log(100) / log(2).
  2. log(100) = 2.
  3. log(2) ≈ 0.3010.
  4. Result: 2 / 0.3010 ≈ 6.644.

The formula works identically with ln in both numerator and denominator. Choose whichever is more convenient.

Logarithmic Equations Calculator: Solve a Logarithmic Equation Step-by-Step

A logarithmic equation is any equation where the variable sits inside a logarithm. The basic strategy is the same every time: isolate the logarithmic expression, then rewrite in exponential form.

What is a logarithmic equation? It is an equation like log₂(x) = 5. To solve, convert to exponential form: x = 2⁵ = 32.

General steps to solve a logarithmic equation:

  1. Combine or simplify logarithmic expressions using the laws of logarithms.
  2. Isolate the log term on one side.
  3. Convert to an exponential equation.
  4. Solve for the variable.
  5. Check your answer. The argument of the logarithm must be positive.

Equations Step-by-Step Solution for Common Logarithmic Equations

Example 1: log(x) = 3

  1. Base is 10 (common log).
  2. Rewrite: x = 10³.
  3. x = 1000.

Example 2: ln(x + 2) = 4

  1. Rewrite: x + 2 = e⁴.
  2. e⁴ ≈ 54.598.
  3. x ≈ 52.598.
  4. Check: 52.598 + 2 = 54.598 > 0. Valid.

Example 3: log₂(x) + log₂(x − 2) = 3

  1. Use the multiplication property: log₂(x(x − 2)) = 3.
  2. Rewrite: x(x − 2) = 2³ = 8.
  3. Expand: x² − 2x − 8 = 0.
  4. Factor: (x − 4)(x + 2) = 0.
  5. Solutions: x = 4 or x = −2.
  6. Discard x = −2 because the argument must be positive.
  7. Answer: x = 4.

Always verify that your solution keeps every argument of the logarithm positive. Negative arguments are undefined in real-number math.

Logarithmic Math: Exponent, Value, and Exponential Relationships

Logarithms and exponents are two sides of the same coin. If you understand one, you understand the other.

What is the inverse of log in math? Exponentiation. If log base b of x = y, then b raised to y = x. They undo each other.

Key properties (laws of logarithms):

  • Product rule: log_b(m × n) = log_b(m) + log_b(n). Multiplication inside becomes addition outside.
  • Quotient rule: log_b(m / n) = log_b(m) − log_b(n). Division inside becomes subtraction outside.
  • Power rule: log_b(m^k) = k × log_b(m). An exponent inside comes out front as a multiplier.
  • Identity: log_b(b) = 1, because b¹ = b.
  • Zero rule: log_b(1) = 0, because b⁰ = 1. This answers the common question: what is log 1? It is always 0, regardless of the base.

These properties let you simplify complex logarithmic expressions before solving. They also explain why logarithms convert multiplication into addition, a property that made them essential for computation long before electronic calculators existed. John Napier introduced logarithms in the early 1600s precisely for this reason.

Using This Log Calculator for Science, Engineering, and Statistics

Logarithms show up in nearly every quantitative field. Here are practical uses where this calculator saves time.

Science and physics:

  • The pH scale measures acidity as pH = −log[H⁺]. A quick base 10 calculation gives you the answer.
  • The Richter scale for earthquake magnitude is logarithmic. Each whole number increase represents a tenfold jump in measured amplitude.
  • Sound intensity (decibels) uses 10 × log(I / I₀).

Engineering and computer science:

  • Algorithm complexity often involves log base 2 (binary search, tree depth).
  • Signal processing uses logarithmic scales for frequency response.

Finance and economics:

  • Continuous compound interest uses ln: A = P × e^(rt). Solving for time requires a natural logarithm.
  • Log returns in economics smooth out percentage changes for comparison.

Statistics:

  • Log transformations normalize skewed data.
  • Logistic regression uses the natural log of odds.

For any of these applications, enter your values, choose the base, and get your result. The tool handles the formula so you can focus on interpretation.

Logarithm Calculation Limitations and When to Seek a Different Solution

Every calculator has boundaries. Understanding them prevents errors.

Can you have a negative log? The result (output) of a logarithm can be negative. For example, log(0.01) = −2. However, the argument (input) must be positive. You cannot take the logarithm of zero or a negative number in real-number math.

The base must be positive and cannot equal 1. log base 1 of any number is undefined because 1 raised to any power is always 1.

Floating-point precision. This calculator provides accurate estimates, but very large or very small numbers may show minor rounding. For mission-critical engineering calculations, use professional-grade software and verify independently.

When to seek a different solution:

  • Complex numbers: if you need logarithms of negative numbers, you need the complex logarithm, which is outside the scope of this tool.
  • Symbolic algebra: if your equation has multiple log terms with different bases and variables, a dedicated computer algebra system may be more appropriate.
  • Professional decisions: this is an estimate and learning aid. Do not use it as the sole basis for engineering sign-offs, medical calculations, or legal conclusions.

For straightforward logarithm calculation across any base, this free log calculator online gives you a fast, reliable answer.