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Logarithm Calculator – Compute Log of Any Base Online

Calculate logarithms of any number for any base with our free online logarithm calculator. Supports log base 10, natural log (ln), and custom base logarithms with instant results.

Examples:

Understanding Logarithms

A logarithm answers the question: "To what power must I raise the base to get this number?" For example, log₁₀(100) = 2 because 10² = 100. Logarithms are the inverse operation of exponentiation, just as division is the inverse of multiplication.

The notation log_b(x) = y means b^y = x. Three bases are especially common: base 10 (common log, used in science and engineering), base e (natural log, used in calculus and growth models), and base 2 (binary log, used in computer science).

Logarithm Properties

Product Rule

log_b(xy) = log_b(x) + log_b(y)

The log of a product equals the sum of the logs

Quotient Rule

log_b(x/y) = log_b(x) - log_b(y)

The log of a quotient equals the difference of the logs

Power Rule

log_b(x^n) = n × log_b(x)

The log of a power brings the exponent down as a multiplier

Change of Base

log_b(x) = log_c(x) / log_c(b)

Convert any log to a different base

Worked Examples

Example 1: Common Logarithm

Find log₁₀(1000)

Question: 10 raised to what power equals 1000?
10¹ = 10
10² = 100
10³ = 1000 ✓
Answer: log₁₀(1000) = 3

Example 2: Natural Logarithm

Find ln(e⁵)

By definition, ln(x) = logₑ(x)
ln(e⁵) asks: e raised to what power equals e⁵?
Answer is clearly 5
Answer: ln(e⁵) = 5
General rule: ln(e^x) = x

Example 3: Binary Logarithm

Find log₂(64)

Question: 2 raised to what power equals 64?
2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32, 2⁶ = 64 ✓
Answer: log₂(64) = 6

Example 4: Custom Base

Find log₃(81)

Question: 3 raised to what power equals 81?
3¹ = 3, 3² = 9, 3³ = 27, 3⁴ = 81 ✓
Or using change of base: log₃(81) = ln(81)/ln(3) = 4.394/1.099 = 4
Answer: log₃(81) = 4

Example 5: Log of a Decimal

Find log₁₀(0.01)

0.01 = 1/100 = 10⁻²
So log₁₀(0.01) = log₁₀(10⁻²) = -2
Answer: log₁₀(0.01) = -2
Logs of numbers between 0 and 1 are negative

Quick Fact

John Napier invented logarithms in 1614 to simplify astronomical calculations. Before calculators, scientists used log tables to turn multiplication into addition. The slide rule, based on logarithmic scales, was the primary calculation tool for engineers until electronic calculators arrived in the 1970s.

Frequently Asked Questions

Why can't I take the log of a negative number?

No real number raised to any power gives a negative result. For example, there's no power you can raise 10 to that equals -100. (In complex numbers, logs of negatives exist, but that's advanced mathematics.)

What's the difference between log and ln?

"log" without a base usually means log₁₀ (base 10). "ln" always means logₑ (base e, where e ≈ 2.718). In higher mathematics, "log" sometimes means natural log – context matters.

What is log(1)?

log_b(1) = 0 for any valid base b. This is because any number raised to the power 0 equals 1. So log₁₀(1) = 0, ln(1) = 0, log₂(1) = 0, etc.

Where are logarithms used in real life?

Logarithms appear everywhere: pH scale (acidity), Richter scale (earthquakes), decibels (sound), musical intervals, compound interest calculations, population growth models, and data compression algorithms.

What's special about base e?

The number e (≈2.718) is the base for natural growth. Functions like e^x have the unique property that their derivative equals themselves. This makes natural logs essential in calculus, physics, and modeling continuous growth or decay.

How do I calculate log without a calculator?

For simple cases, think about powers: log₁₀(1000) = 3 because 10³ = 1000. For other values, use log tables or the change-of-base formula with known values. Memorize key values like log₁₀(2) ≈ 0.301 and log₁₀(3) ≈ 0.477.

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