CalculatorForYou.online Logo CalculatorForYou.online
About Advertising Affiliate Disclosure Contact Cookie Policy Disclaimer GDPR Compliance Privacy Policy Scientific Calculator Guide Sitemap Terms of Service

📊 Logarithm Calculator (Log, Ln, Log10, Log2)

Quickly calculate logarithms and antilogs with step-by-step working

This free logarithm calculator helps you work with logs the way you actually use them in class, exams, and real life. You can calculate log₁₀, ln, log₂, custom base logarithms, and antilogs in one place. Each result comes with the main formula, a short step-by-step explanation, and a quick verification so you can see how the answer was formed. For related exponential work, you can also use our Exponent Calculator and Scientific Calculator.

📐 Common Logarithm (log₁₀)

Base-10 logarithms are used heavily in science and engineering. They appear in the Richter scale, sound levels in decibels, and many formulas for compound interest and growth.

🌿 Natural Logarithm (ln)

Natural logs (base e) are key in calculus, exponential growth and decay, and continuous compounding. They pair perfectly with tools like our Compound Interest Calculator.

💻 Binary Logarithm (log₂)

Binary logs are the default in computer science. Use log₂ to understand algorithm complexity, binary trees, and information in bits alongside our Basic Calculator for quick checks.

⚙️ Custom Base & Antilog

Use a custom base for any log you need and switch to Antilog mode to recover the original number. This is helpful when working with tables, slide-rule style calculations, or exam questions involving antilogs.

Logarithms Explained: How Log, Ln & Log10 Work (With Real-Life Examples)

Logarithms look scary at first, but they’re really just a smart way to undo exponents. Whenever a quantity grows or shrinks exponentially, logs help you “bring the power down” and solve for time, rate, or size. You’ll see them in school algebra, university calculus, physics, chemistry, finance, data science, and computer science. If you are revising powers and exponents, try this calculator together with our Exponent Calculator and Percentage Calculator for a complete practice set.

What Are Logarithms? Understanding the Fundamental Concept

A logarithm answers the question: “To what power must we raise a base to get a certain number?” In mathematical notation, if bx = y, then logb(y) = x. The logarithm is the inverse operation of exponentiation, just as division is the inverse of multiplication and subtraction is the inverse of addition.

Basic Logarithm Definition: If b^x = y, then log_b(y) = x Components: - b = base (must be positive, not equal to 1) - y = argument (must be positive) - x = result (the logarithm) Examples: log₁₀(100) = 2 because 10² = 100 log₂(8) = 3 because 2³ = 8 log₅(125) = 3 because 5³ = 125 ln(e) = 1 because e¹ = e Reading Logarithms: log₁₀(100) is read as "log base 10 of 100" ln(x) is read as "natural log of x"

The logarithm–exponent relationship: Logarithms and exponents are inverse operations. If you take the logarithm of an exponential expression with the same base, you get back the exponent: logb(bx) = x. Similarly, if you exponentiate a logarithm with the same base, you get back the argument: blogb(x) = x.

Types of Logarithms: Common, Natural, and Binary

Common Logarithm (log₁₀)

The common logarithm uses base 10 and is written as log₁₀(x) or simply log(x). Before calculators, common logarithms were essential for scientific calculations because our number system is base 10. They’re still widely used in science and engineering.

Common Logarithm Examples: log₁₀(10) = 1 log₁₀(100) = 2 log₁₀(1000) = 3 log₁₀(1) = 0 log₁₀(0.1) = -1 log₁₀(0.01) = -2 Pattern: log₁₀(10^n) = n For any power of 10, the log is simply the exponent.

Where log₁₀ shows up in real life: The Richter scale for earthquakes uses log₁₀ – each whole number increase represents a tenfold increase in amplitude. Sound intensity in decibels also uses log₁₀: dB = 10 × log₁₀(I/I₀). pH in chemistry is defined as pH = −log₁₀[H⁺].

Natural Logarithm (ln)

The natural logarithm uses base e (Euler’s number, approximately 2.71828) and is written as ln(x) or loge(x). Natural logarithms are fundamental in calculus because the derivative of ln(x) is 1/x, making them essential for solving differential equations and continuous growth problems.

Natural Logarithm Examples: ln(e) = 1 ln(e²) = 2 ln(1) = 0 ln(e⁻¹) = -1 Special Properties: d/dx [ln(x)] = 1/x ∫(1/x) dx = ln|x| + C Euler's Number: e ≈ 2.718281828... e = lim(n→∞) (1 + 1/n)^n

Why natural logs feel “natural”: Natural logarithms appear automatically in continuous growth and decay processes. Continuous compound interest uses e in the formula A = Pert. Population growth, radioactive decay, and many physical phenomena follow exponential patterns with base e, making ln the natural choice for analysis. Our Compound Interest Calculator is a good place to see this in action.

Binary Logarithm (log₂)

The binary logarithm uses base 2 and is written as log₂(x). Binary logarithms are essential in computer science and information theory because computers work in base 2.

Binary Logarithm Examples: log₂(2) = 1 log₂(4) = 2 log₂(8) = 3 log₂(16) = 4 log₂(1024) = 10 Computer Science Uses: - Number of bits needed: ⌈log₂(n)⌉ - Binary search complexity: O(log₂ n) - Tree height: log₂(nodes) - Information entropy: −Σ p(x) log₂ p(x)

In information theory, the information content of an event with probability p is −log₂(p) bits. A fair coin flip (p = 0.5) carries 1 bit of information.

Logarithm Laws and Properties: Essential Rules

The laws of logarithms let you simplify expressions and solve equations more easily. They follow directly from the laws of exponents.

Fundamental Laws of Logarithms: 1. Product Rule: log_b(xy) = log_b(x) + log_b(y) 2. Quotient Rule: log_b(x/y) = log_b(x) − log_b(y) 3. Power Rule: log_b(x^n) = n × log_b(x) 4. Change of Base: log_b(x) = log_a(x) / log_a(b) 5. Identity: log_b(b) = 1 6. Zero: log_b(1) = 0 7. Inverse Property: b^(log_b(x)) = x log_b(b^x) = x

Why the product rule works: If bm = x and bn = y, then xy = bm+n, so logb(xy) = m + n = logb(x) + logb(y). The other rules follow the same logic using exponential properties.

Change of Base Formula: Converting Between Log Bases

The change of base formula is what your calculator uses behind the scenes for custom bases. Most calculators only have ln and log, but you can still compute any base: logb(x) = ln(x)/ln(b) or log(x)/log(b).

Change of Base Formula: log_b(x) = log_a(x) / log_a(b) Common versions: log_b(x) = ln(x) / ln(b) log_b(x) = log₁₀(x) / log₁₀(b) Example: log₂(10) = ln(10) / ln(2) ≈ 2.303 / 0.693 ≈ 3.322

Antilogarithm: Reversing a Log

The antilogarithm (or antilog) simply reverses a logarithm. If logb(x) = y, then antilogb(y) = x = by. In the calculator, Antilog mode does this step for you and then checks the result by taking the log again.

Antilogarithm Definition: If log_b(x) = y, then antilog_b(y) = x = b^y Examples: antilog₁₀(2) = 10² = 100 antilog₁₀(3) = 10³ = 1000 antilog_e(1) = e¹ ≈ 2.718 antilog₂(5) = 2⁵ = 32

Real-World Applications of Logarithms

You’ll find logarithms in many topics covered by other tools on CalculatorForYou.online — from mortgage payments to retirement planning and even health metrics when growth and decay models are involved.

Logarithm Calculator – Frequently Asked Questions

1. Can I use this logarithm calculator for homework and exams?

Yes. The calculator is designed to match what you learn in class, with clear steps and formulas. Always check your teacher’s rules about calculator use in exams, but for practice and homework it’s a great companion.

2. Does the calculator handle log of decimal and fractional numbers?

Yes. As long as the input is positive, you can enter whole numbers, decimals, or fractions (in decimal form) and the calculator will compute the log or antilog accurately.

3. What should I do if the calculator says the input is not valid?

Logarithms are only defined for positive arguments, and the base must be positive and not equal to 1. If you see an error, make sure both the base and the value follow these rules.

4. Which mode should I choose: log, ln, log₂, or custom base?

Use log (log₁₀) for common scientific scales, ln for calculus and continuous growth, log₂ for computer science, and Custom Base when a question gives you a specific base like log₇ or log₀.5.

5. How is this different from a normal scientific calculator?

A scientific calculator gives you the final number. This logarithm calculator also shows the key formula, a short step-by-step breakdown, and a verification so you can understand what is happening — not just copy the result.

6. Which other tools should I use with this calculator?

For a complete study set, combine this page with our Exponent Calculator, Percentage Calculator, and Scientific Calculator. Together they cover most exponential and logarithmic questions you’ll see in school and college.

Conclusion: Make Logarithms Less Scary and More Useful

Logarithms are everywhere once you start noticing them — in earthquake scales, sound levels, pH, compound interest, population growth, and algorithm complexity. They turn messy exponential relationships into manageable linear ones. When you combine this logarithm calculator with tools like the Exponent Calculator and Compound Interest Calculator, you get a complete mini-toolbox for tackling exam questions and real-world problems.

Use this page to check answers, learn the patterns, and build intuition. The more you practice, the more natural log, ln, log₁₀, and log₂ will feel — and the less intimidating exponential questions will become.

🚀 Ready to try it yourself? Use the logarithm calculator at the top of this page to explore log, ln, log₁₀, log₂, custom base logs, and antilogs. Change the mode, tweak the inputs, and watch how the steps and explanations update in real time.