A logarithm answers the question: 'To what power must the base be raised to get this number?' For example, log₁₀(100) = 2 because 10² = 100. The logarithm is the inverse operation of exponentiation.

What is the difference between ln, log, log₁₀, and log₂?

ln (natural log) uses base e ≈ 2.71828. log₁₀ (common log) uses base 10 and is often written as just 'log'. log₂ (binary log) uses base 2, commonly used in computer science. 'log' with a custom base can be any positive number except 1.

How do I calculate log with a different base?

Use the change of base formula: logₐ(x) = ln(x) / ln(a) = log₁₀(x) / log₁₀(a). This calculator automatically applies this formula for any base you choose.

Antilog is the inverse of logarithm. If log_b(x) = y, then antilog_b(y) = x = b^y. For example, antilog₁₀(2) = 10² = 100. It converts a logarithmic value back to the original number.

Why can't I calculate log of a negative number or zero?

In real numbers, logarithms are only defined for positive values. The log of zero approaches negative infinity, and logs of negative numbers require complex numbers. This calculator works with real numbers only.

What are common uses of logarithms?

Logarithms are used in: measuring earthquake intensity (Richter scale), sound levels (decibels), pH in chemistry, compound interest calculations, algorithm complexity in computer science (O(log n)), and data compression.

How do I solve a logarithmic equation step by step?

First, use the log rules to condense each side into a single logarithm. If you have log_b(M) = log_b(N), set M = N (the one-to-one property). If you have a log on one side and a number on the other, like log_b(x) = y, rewrite it in exponential form: x = b^y. Always check your answer in the original equation and reject any solution that makes a log argument zero or negative (an extraneous solution). Example: log₂(x) + log₂(x−2) = 3 condenses to log₂(x²−2x) = 3, so x²−2x = 8, giving x = 4 (x = −2 is rejected).

What are the laws of logarithms?

There are three core laws: the product rule log_b(MN) = log_b(M) + log_b(N), the quotient rule log_b(M/N) = log_b(M) − log_b(N), and the power rule log_b(Mᵏ) = k·log_b(M). Two identities follow: log_b(1) = 0 and log_b(b) = 1. These laws let you expand a single logarithm into a sum or condense several logs into one.

How do I expand and condense logarithmic expressions?

To expand, apply the laws left to right: log_b(x²y/z) becomes 2·log_b(x) + log_b(y) − log_b(z). To condense, work in reverse — turn coefficients into exponents with the power rule, then combine sums into products and differences into quotients. Condensing to a single log is usually the first step in solving a logarithmic equation.

How do I solve for an exponent using logarithms?

Take the log of both sides. To solve 2^x = 50, take log₂ of both sides: x = log₂(50) = ln(50)/ln(2) ≈ 5.644. Any base works with the change-of-base formula, so you can use the ln or log₁₀ button on any calculator to evaluate it.

How do I change the base of a logarithm?

Use the change-of-base formula: log_b(x) = log_c(x) / log_c(b), where c is any base your calculator supports (usually e or 10). For example, log₂(100) = ln(100)/ln(2) = log₁₀(100)/log₁₀(2) ≈ 6.644. This calculator applies the formula automatically for any base you enter.

What is an extraneous solution in a logarithmic equation?

An extraneous solution is a value that appears during algebra but is invalid because it makes a logarithm's argument zero or negative — which is undefined for real logs. After solving, substitute every candidate back into the original equation and discard any that produce log of a non-positive number.

When should I use ln versus log₁₀?

Use natural log (ln, base e) for continuous growth and decay, calculus, and anything involving the constant e — such as continuously compounded interest or radioactive decay. Use common log (log₁₀) for orders of magnitude and scales built on powers of ten, such as the Richter scale, decibels, and pH. Mathematically they differ only by a constant factor: ln(x) = log₁₀(x) × ln(10).

Why is log₂ important in computer science?

Binary log (log₂) counts how many times you can halve a number before reaching 1, which is exactly the work done by divide-and-conquer algorithms. A binary search over n = 1,000,000 items takes about log₂(1,000,000) ≈ 20 comparisons. Log₂ also measures information in bits and appears throughout Big O complexity analysis, like O(n log n) sorting.

Logarithm Calculator

Logarithms convert multiplicative relationships into additive ones, making them essential in acoustics (decibels), seismology (Richter scale), chemistry (pH), information theory (bits and entropy), and financial modeling (continuously compounded interest). This calculator evaluates log in any base, natural log (ln), and log₂ — and supports the change-of-base formula for converting between bases. Whether you are solving exponential equations, analyzing signal strength, or studying algorithmic complexity (Big O notation), logarithms are the tool of choice.

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functions Logarithm Calculator

Calculation Type

Value (x)

Must be a positive number

Base (b)

Must be positive and not equal to 1

Decimal Places

Log Rules

log(a×b) = log(a) + log(b)

log(a/b) = log(a) - log(b)

log(aⁿ) = n × log(a)

calculate Result

Expression

log₁₀(100)

Result

Verification

10² = 100 ✓

Same Value in Other Bases

ln (base e) 4.605170

log₁₀ 2.000000

log₂ 6.643856

Change of Base Formula

log_b(x) = ln(x) / ln(b)

Input Values

Calculation Type

expand_more

Value (x)

Base (b)

Decimal Places

expand_more

lightbulb Tips

•log_b(x) = y means b^y = x
•ln uses base e ≈ 2.718
•log(ab) = log(a) + log(b)
•log(a^n) = n × log(a)

functions Reference

Common Logs

log₁₀(10) = 1log₁₀(100) = 2 log₂(8) = 3log₂(16) = 4 ln(e) = 1ln(1) = 0

Constants

e ≈ 2.71828182845904

How to Use the Logarithm Calculator

calculate

Choose Type

Select log with base, natural log (ln), log₁₀, log₂, or antilog.

edit

Enter Value

Input the number to calculate the logarithm of.

tune

Set Base

For custom log, enter the base (default: 10).

visibility

View Results

See the result with verification and steps.

The Formula

A logarithm answers: 'To what power must the base be raised to get this number?' It's the inverse of exponentiation.

log_b(x) = y means b^y = x

lightbulb Variables Explained

log_b(x) Logarithm of x with base b
ln(x) Natural logarithm (base e)
log(x) Common logarithm (base 10)

tips_and_updates Pro Tips

log₁₀(x) is called 'common log' and often written as just 'log'

ln(x) is natural log with base e ≈ 2.71828

log₂(x) is binary log, commonly used in computer science

Logarithm is the inverse of exponentiation: if bˣ = y, then log_b(y) = x

log(a×b) = log(a) + log(b) - useful for simplifying calculations

Change of base: log_a(x) = ln(x)/ln(a) = log(x)/log(a)

Calculate logarithms with any base, natural log (ln), common log (log₁₀), binary log (log₂), and antilog. Get step-by-step solutions, solve logarithmic equations, and review the log rules.

Understanding Logarithms: The Inverse of Exponentiation

A logarithm answers one question: to what power must the base be raised to produce a given number? If 10² = 100, then log₁₀(100) = 2. Because the logarithm undoes exponentiation, it compresses huge ranges of values into manageable numbers — which is why it underpins scientific scales, signal processing, and algorithm analysis. The base must be positive and not equal to 1, and the argument must be positive, since no power of a positive base yields zero or a negative result.

Types of Logarithms: Log₁₀, Natural Log (ln), and Log₂

Common log (log₁₀) uses base 10 and is the default when 'log' is written with no base — it measures orders of magnitude on scales like the Richter scale and decibels. Natural log (ln) uses base e ≈ 2.71828 and is the language of continuous growth, calculus, and compound interest. Binary log (log₂) uses base 2 and counts halvings, making it central to computer science and information theory. All three are connected by the change-of-base formula, so any one can be computed from another.

How to Solve Logarithmic Equations Step by Step

Solving a logarithmic equation follows a repeatable process. First, use the log rules to condense each side into a single logarithm. If both sides are logs with the same base — log_b(M) = log_b(N) — apply the one-to-one property and set M = N. If a log equals a plain number, log_b(x) = y, rewrite it in exponential form as x = b^y. For example, log₂(x) + log₂(x−2) = 3 condenses to log₂(x²−2x) = 3, so x²−2x = 2³ = 8, which factors to (x−4)(x+2) = 0, giving x = 4 or x = −2. Always substitute back: x = −2 makes log₂(−2) undefined, so it is an extraneous solution and is rejected, leaving x = 4.

Logarithm Rules and Laws: Product, Quotient, and Power

Three laws govern every logarithm. The product rule states log_b(MN) = log_b(M) + log_b(N); the quotient rule states log_b(M/N) = log_b(M) − log_b(N); and the power rule states log_b(Mᵏ) = k·log_b(M). Two identities follow directly: log_b(1) = 0 and log_b(b) = 1. To change base, use log_b(x) = ln(x)/ln(b) = log₁₀(x)/log₁₀(b). These laws let you expand a single log into a sum — log_b(x²y/z) = 2·log_b(x) + log_b(y) − log_b(z) — or condense several logs into one, which is the essential first step when solving an equation or simplifying an expression.

Real-World Applications: Richter Scale, Decibels, and pH

Logarithms turn multiplication into addition, which is why they describe scales that span many orders of magnitude. A magnitude-6 earthquake releases 10² = 100 times the ground-shaking amplitude of a magnitude-4 quake, because the Richter scale is log₁₀ of amplitude. Sound is measured in decibels as 10·log₁₀(P/P₀), and acidity in chemistry as pH = −log₁₀[H⁺]. In finance, the time to grow an investment under continuous compounding is t = ln(A/P)/r. In computing, binary search runs in O(log₂ n) time — about 20 steps to find one item among a million.

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Frequently Asked Questions

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Logarithm Calculator

lightbulb Tips

functions Reference

How to Use the Logarithm Calculator

Choose Type

Enter Value

Set Base

View Results

The Formula

lightbulb Variables Explained

tips_and_updates Pro Tips

Understanding Logarithms: The Inverse of Exponentiation

Types of Logarithms: Log₁₀, Natural Log (ln), and Log₂

How to Solve Logarithmic Equations Step by Step

Logarithm Rules and Laws: Product, Quotient, and Power

Real-World Applications: Richter Scale, Decibels, and pH

Embed this Logarithm Calculator on your site

Frequently Asked Questions

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