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.

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Logarithm Calculator

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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