What is a logarithm and how do you calculate it?

A logarithm answers the question: to what power must I raise the base to get this number? The notation log_b(x) = y means b^y = x. For example, log₂(8) = 3 because 2³ = 8. The three components are the base (b), the argument (x), and the result (y — the logarithm). To calculate: use the change-of-base formula log_b(x) = ln(x)/ln(b), which works for any base. The three most common logarithms: • log₁₀(x): common logarithm (base 10) — used in engineering, pH, decibels, Richter scale • ln(x): natural logarithm (base e ≈ 2.71828) — used in calculus, physics, finance (continuous compounding) • log₂(x): binary logarithm (base 2) — used in computer science, information theory • All three are available in this calculator via the mode selector

What is the difference between ln and log?

ln(x) is the natural logarithm — it uses the mathematical constant e (≈ 2.71828) as its base. log(x) usually refers to the common logarithm with base 10, though in higher mathematics 'log' often means 'ln'. The relationship is: ln(x) = log₁₀(x) / log₁₀(e) = log₁₀(x) / 0.43429, or equivalently log₁₀(x) = ln(x) × 0.43429. Where each is used: • ln(x): calculus derivatives (d/dx[ln x] = 1/x), continuous compound interest (A = Pe^(rt)), physics (radioactive decay, entropy), natural growth and decay • log₁₀(x): pH = −log₁₀[H⁺], decibels = 10×log₁₀(P₂/P₁), Richter scale, mantissa in logarithm tables • log₂(x): bits of information = log₂(number of outcomes), algorithm complexity (binary search = O(log₂ n)) • They are all proportional — ln(x) = log₁₀(x) × ln(10) ≈ log₁₀(x) × 2.302585

What is the change of base formula?

The change of base formula allows you to compute a logarithm in any base using a calculator that only has ln or log₁₀. The formula is: log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b). For example, log₂(100) = ln(100)/ln(2) = 4.60517/0.69315 = 6.64386. Why it works: • If log_b(x) = y, then b^y = x • Take ln of both sides: ln(b^y) = ln(x) • Using log power rule: y × ln(b) = ln(x) • Solve for y: y = ln(x)/ln(b) = log_b(x) ✓ • The same derivation works with log₁₀ instead of ln • This formula is what this calculator uses internally for all log computations • Key insight: all logarithms of the same argument are proportional to each other — they differ only by a constant multiplier

What are the laws of logarithms?

The laws of logarithms are rules for simplifying log expressions. They mirror the laws of exponents, since logarithms are the inverse of exponentials. The six essential log laws: • Product rule: log_b(xy) = log_b(x) + log_b(y) — log of a product = sum of logs • Quotient rule: log_b(x/y) = log_b(x) − log_b(y) — log of a quotient = difference of logs • Power rule: log_b(xⁿ) = n × log_b(x) — log of a power = exponent times log • Identity: log_b(b) = 1 — log of the base equals 1 • Zero: log_b(1) = 0 — log of 1 is always 0 for any base • Reciprocal: log_b(1/x) = −log_b(x) Common applications: • Product rule was used in pre-calculator era to multiply large numbers using log tables • Power rule makes logarithms useful for solving exponential equations • These laws apply identically to ln and log₁₀ — the base does not affect the form

Why is the logarithm undefined for x ≤ 0?

The logarithm log_b(x) asks: what power of b gives x? For positive bases b > 1, the expression b^y is always strictly positive for any real y — it approaches 0 but never reaches it, and it is never negative. So there is no real number y such that b^y = 0 or b^y = x for x 0. Intuitive explanation: • 2^y is always positive: 2^0 = 1, 2^0.001 ≈ 1.0007, 2^−100 ≈ 10⁻³⁰ (tiny but positive) • No power of 2 ever equals 0 or goes negative • log₂(0) would mean 2^? = 0 — impossible for real y • log₂(−1) would mean 2^? = −1 — impossible for real y (complex logarithms exist but use imaginary numbers) • The logarithm curve approaches −∞ as x → 0⁺, but x = 0 is a vertical asymptote • This calculator shows an error message when x ≤ 0 is entered

Logarithm Calculator — Any Base, ln & log₁₀

The Logarithm Calculator computes log_b(x) for any base, ln(x) (natural logarithm, base e), and log₁₀(x) (common logarithm, base 10). Select your mode, enter the value and (for custom base) the base — the result is shown instantly using the change-of-base formula ln(x)/ln(b). The step-by-step soluti...

LOGARITHM CALCULATOR

log_2(8) = 3

Base (b)

Value (x) — must be > 0

log_2(8) =

CHANGE OF BASE (ln)

ln(8) / ln(2) = 2.079442 / 0.693147

INVERSE CHECK

2^3 = 8 ≈ 8 ✓

STEP-BY-STEP

Question: log_2(8) = ?

This asks: 2 raised to what power equals 8?

Using change-of-base formula: log_b(x) = ln(x) / ln(b)

= 2.079442 / 0.693147

= 3

Verify: 2^3 = 8 ≈ 8 ✓

QUICK EXAMPLES

LOG CURVE

y = log_2(x)

INPUT (x)

RESULT

INVERSE

2^3

Red dot = your input · Curve passes through (1, 0) always

COMMON VALUES — BASE 2

log(0.001)-9.9658

log(0.01)-6.6439

log(0.1)-3.3219

log(1)0

log(2)1

log(e)1.4427

log(5)2.3219

log(10)3.3219

Created with❤️byeaglecalculator.com

LAWS OF LOGARITHMS

APPLIED WITH x = 8, BASE = 2

LAW	EXAMPLE	RESULT
log_b(xy) = log_b(x) + log_b(y)	log_2(8 × 2)	3 + 1 = 4
log_b(x/y) = log_b(x) − log_b(y)	log_2(8 / 2)	3 − 1 = 2
log_b(xⁿ) = n × log_b(x)	log_2(8²)	2 × 3 = 6
log_b(1) = 0	log_2(1)	0
log_b(b) = 1	log_2(2)	1
log_b(1/x) = −log_b(x)	log_2(1/8)	−3 = -3

HOW TO USE

1
Select your mode: logᵦ(x) for any custom base, ln(x) for the natural logarithm (base e), or log₁₀(x) for the common logarithm (base 10).
2
For custom base mode: enter the base b (must be greater than 0 and not equal to 1). Enter the value x — it must be strictly greater than 0.
3
The result appears instantly as log_b(x). The change-of-base representation shows how the result is computed as ln(x)/ln(b).
4
Check the inverse verification — the calculator confirms that b^result ≈ x, proving the answer is correct.
5
Review the Laws of Logarithms table — each law is illustrated with your specific x and base, making the rules concrete and memorable.

WORKED EXAMPLE

Example 1: log₂(8). base=2, x=8. Result = ln(8)/ln(2) = 2.07944/0.69315 = 3. Verify: 2³ = 8 ✓. Example 2: log₁₀(1000) = ln(1000)/ln(10) = 6.90776/2.30259 = 3. Verify: 10³ = 1000 ✓. Example 3: ln(e) = 1. Verify: e¹ = e ✓. Example 4: log₂(1/8) = log₂(0.125) = ln(0.125)/ln(2) = −2.07944/0.69315 = −3. Verify: 2⁻³ = 1/8 ✓.

REFERENCE FORMULAS

LOGARITHM LAWS REFERENCE

NAME	FORMULA	DESCRIPTION
Definition	log_b(x) = y ↔ bʸ = x	Logarithm is the inverse of exponentiation
Change of base	log_b(x) = ln(x) / ln(b)	Convert any base using natural log
Product rule	log_b(xy) = log_b(x) + log_b(y)	Log of product = sum of logs
Quotient rule	log_b(x/y) = log_b(x) − log_b(y)	Log of quotient = difference of logs
Power rule	log_b(xⁿ) = n × log_b(x)	Log of power = exponent × log
Identity	log_b(b) = 1, log_b(1) = 0	Log of the base = 1, log of 1 = 0
ln vs log₁₀	ln(x) = log₁₀(x) × ln(10) ≈ 2.3026 × log₁₀(x)	Convert between natural and common log

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Last updated: April 28, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.