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Logarithm Calculator
Enter any number and choose a base - log₁₀, natural log (ln), log₂, or any custom base - to get the exact answer with a full step-by-step solution. Includes antilogarithm, change of base conversion between any two bases, all logarithm laws with worked examples, and an interactive graph.
Logarithm Calculator
log₁₀(x) = ?
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Logarithm Reference Table
Click any row to calculate. Values rounded to 6 decimal places.
| x | log₁₀(x) | ln(x) | log₂(x) | √x |
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Logarithm Graph
log₁₀(x)
ln(x)
log₂(x)
Logarithm Rules & Properties
Definition
logₙ(x) = y means nʸ = x. The logarithm is the inverse of the exponential.
log₁₀(1000) = 3 because 10³ = 1000
ln(e²) = 2 because e² = e²
log₂(8) = 3 because 2³ = 8
ln(e²) = 2 because e² = e²
log₂(8) = 3 because 2³ = 8
Product Rule
log(a × b) = log(a) + log(b)
Example: log(6) = log(2×3) = log(2) + log(3)
= 0.3010 + 0.4771 = 0.7781
Example: log(6) = log(2×3) = log(2) + log(3)
= 0.3010 + 0.4771 = 0.7781
Quotient Rule
log(a / b) = log(a) − log(b)
Example: log(5) = log(10/2) = log(10) − log(2)
= 1 − 0.3010 = 0.6990
Example: log(5) = log(10/2) = log(10) − log(2)
= 1 − 0.3010 = 0.6990
Power Rule
log(aⁿ) = n × log(a)
Example: log(1000) = log(10³) = 3 × log(10) = 3
Example: log(1000) = log(10³) = 3 × log(10) = 3
Change of Base Formula
Convert between any logarithm bases.
logₙ(x) = log(x) / log(n) = ln(x) / ln(n)
Example: log₅(25) = log(25)/log(5) = 1.3979/0.6990 = 2
Verify: 5² = 25
Example: log₅(25) = log(25)/log(5) = 1.3979/0.6990 = 2
Verify: 5² = 25
Special Values
log(1) = 0 (any base, because b⁰ = 1)
log(b) = 1 (log of the base itself = 1)
log(0) = undefined (−∞ approaching)
log(−x) = undefined (in real numbers)
ln(e) = 1 (e ≈ 2.71828...)
ln(1) = 0
log₂(2) = 1
log(b) = 1 (log of the base itself = 1)
log(0) = undefined (−∞ approaching)
log(−x) = undefined (in real numbers)
ln(e) = 1 (e ≈ 2.71828...)
ln(1) = 0
log₂(2) = 1
Antilogarithm
Antilog₁₀(x) = 10ˣ
Antilog(ln) = eˣ
Antilog₂(x) = 2ˣ
Example: antilog₁₀(3) = 10³ = 1000
If log(x) = 2.5, then x = 10^2.5 = 316.23
Antilog(ln) = eˣ
Antilog₂(x) = 2ˣ
Example: antilog₁₀(3) = 10³ = 1000
If log(x) = 2.5, then x = 10^2.5 = 316.23
Real-World Applications
Richter Scale: M = log₁₀(A/A₀) (earthquake)
Decibels: dB = 10×log₁₀(P₁/P₀) (sound)
pH scale: pH = −log₁₀([H⁺]) (acidity)
Compound int: t = ln(A/P) / (r) (finance)
Binary search: steps = log₂(n) (computing)
Info theory: bits = log₂(1/p) (entropy)
Decibels: dB = 10×log₁₀(P₁/P₀) (sound)
pH scale: pH = −log₁₀([H⁺]) (acidity)
Compound int: t = ln(A/P) / (r) (finance)
Binary search: steps = log₂(n) (computing)
Info theory: bits = log₂(1/p) (entropy)
Frequently Asked Questions
A logarithm answers: 'to what power must the base be raised to get this number?' The notation log_b(x) = y means b^y = x. Examples: log₁₀(1000) = 3 because 10³ = 1000. log₂(8) = 3 because 2³ = 8. ln(e²) = 2 because e² = e². A logarithm is the inverse of exponentiation - just as division undoes multiplication, logarithm undoes raising to a power. The log of a number represents the exponent in the original power relationship.
log (log₁₀): base 10 - the common logarithm. Written as 'log' without subscript in most science and engineering contexts. Used in pH (chemistry), decibels (acoustics), Richter scale (seismology). ln: natural logarithm, base e ≈ 2.71828. Fundamental in calculus (derivative of ln(x) = 1/x) and continuous growth/decay models. log₂: base 2 - essential in computer science and information theory. They are all equivalent, just different scales: ln(x) = log₁₀(x) ÷ log₁₀(e) ≈ 2.3026 × log₁₀(x).
log_b(x) = log(x) ÷ log(b) = ln(x) ÷ ln(b). This lets you calculate any logarithm using just log₁₀ or ln - which is why scientific calculators only need two log buttons. Example: log₅(125). Using log₁₀: log(125) ÷ log(5) = 2.09691 ÷ 0.69897 = 3. Verify: 5³ = 125 ✓. Another example: log₇(2401). log(2401) ÷ log(7) = 3.38021 ÷ 0.84510 = 4. Verify: 7⁴ = 2401 ✓.
No real number power of any positive base gives exactly 0. For example, 10^x approaches 0 as x→−∞ but never reaches 0. So there is no real value y such that 10^y = 0. From a calculus perspective, log(x) approaches −∞ as x approaches 0 from the positive side - the limit exists but is not a finite number. Additionally, log of negative numbers is also undefined in real numbers - no real power of a positive base produces a negative result (these require complex logarithms).
The antilogarithm is the inverse of the logarithm - it 'undoes' the log. If log₁₀(x) = y, then antilog(y) = 10^y = x. Examples: antilog₁₀(3) = 10³ = 1000. antilog₁₀(2.5) = 10^2.5 ≈ 316.23. For natural log: if ln(x) = y, then antilog_e(y) = e^y. For base 2: if log₂(x) = y, then antilog₂(y) = 2^y. Antilogarithm is used to reverse calculations done in logarithmic form - very common in scientific data analysis where values are recorded as logs.
Because when you multiply numbers with the same base, you add exponents: b^m × b^n = b^(m+n). If a = b^m and c = b^n, then a×c = b^(m+n). Taking log_b of both sides: log_b(a×c) = m + n = log_b(a) + log_b(c). This property was historically revolutionary - it converted multiplication (hard) into addition (easy). Before calculators, engineers used logarithm tables and slide rules based on this property to multiply large numbers.
e ≈ 2.71828182845... is the base of natural logarithms. It appears naturally wherever continuous growth or decay occurs. Its most elegant definition: if you invest $1 at 100% annual interest compounded continuously, after 1 year you have exactly $e. Mathematically: e = limit as n→∞ of (1 + 1/n)^n. The derivative of e^x is e^x (unchanged) - making it the unique base where exponential and logarithm functions have the simplest derivatives. As fundamental to mathematics as π.
log₂(n) tells you how many binary decisions (bits) are needed to identify one of n possibilities. Bits to represent n values = ⌈log₂(n)⌉. A byte (8 bits) represents 2⁸ = 256 values because log₂(256) = 8. Binary search of n items takes at most ⌈log₂(n)⌉ steps - for 1 million items, that is only 20 comparisons. Hash table, balanced tree, and heap operations are O(log n) - meaning log₂(n) operations - so a database with 1 billion records needs roughly 30 comparisons. This is why logarithms are central to algorithm efficiency analysis.
Logarithm Calculator - log₁₀, ln, log₂ and Every Log Law Explained
Logarithms are the inverse of exponentiation - and once that simple idea clicks, everything else about them follows naturally. Every logarithm question asks: "what power produces this result?" Every log law is a consequence of how powers behave. This calculator computes any logarithm instantly and shows the reasoning behind each step.
Core definition: log_b(x) = y means b^y = x. Examples: log₁₀(10,000) = 4 (because 10⁴ = 10,000). ln(e²) = 2 (because e² = e²). log₂(32) = 5 (because 2⁵ = 32). log₅(1) = 0 (because 5⁰ = 1 - always true for any base).
The Three Most Important Logarithm Bases
log₁₀ - Common Logarithm
- Base 10 - written as "log" without subscript in most contexts
- pH scale: pH = −log₁₀[H⁺] (acid/base chemistry)
- Decibel scale: dB = 10 × log₁₀(P₁/P₂) (sound intensity)
- Richter scale: each step = 10× earthquake energy
- Historically used for manual calculation (log tables)
- log₁₀(10) = 1, log₁₀(100) = 2, log₁₀(1000) = 3
ln - Natural Logarithm (base e)
- Base e ≈ 2.71828 (Euler's number)
- The "natural" base - appears in continuous growth and decay
- Continuous compound interest: A = Pe^(rt)
- Fundamental in calculus: d/dx[ln(x)] = 1/x
- Probability distributions (normal, exponential)
- ln(e) = 1, ln(e²) = 2, ln(1) = 0 for all log bases
The Four Core Logarithm Laws - With Proofs
Every log law follows from how exponents work. Understanding the connection makes the rules memorable rather than arbitrary:
- Product Rule: log(a × b) = log(a) + log(b) - Because when you multiply powers, you add exponents: 10^m × 10^n = 10^(m+n). Taking log of both sides gives log(a×b) = log(a) + log(b). This is why logarithms turned multiplication into addition - dramatically simplifying calculation before calculators existed.
- Quotient Rule: log(a ÷ b) = log(a) − log(b) - Dividing powers subtracts exponents: 10^m ÷ 10^n = 10^(m−n).
- Power Rule: log(aⁿ) = n × log(a) - Taking something to the nth power is multiplying n copies: log(a×a×...×a) = n × log(a) by the product rule.
- Change of Base: log_b(x) = log(x) ÷ log(b) = ln(x) ÷ ln(b) - Lets you compute any logarithm using only log₁₀ or ln. This is the formula used by scientific calculators, which only have log and ln keys.
Special Values Every Student Should Know
- log_b(1) = 0 - For any base: anything raised to the power 0 equals 1. So the log of 1 is always 0, regardless of base.
- log_b(b) = 1 - The log of the base itself is always 1 (b¹ = b). log₁₀(10) = 1, ln(e) = 1, log₂(2) = 1.
- log_b(b^n) = n - Log and exponent cancel: log₂(2⁵) = 5, log₁₀(10³) = 3. These are inverse operations.
- b^(log_b(x)) = x - Raising the base to a log also cancels: 10^(log₁₀(37)) = 37.
- log(0) is undefined - No real power of any base gives exactly 0. The log function approaches −∞ as x approaches 0 from the positive side.
- log of negative numbers is undefined (for real numbers) - No real power of a positive base gives a negative number. Negative logarithms require complex numbers.
Logarithms in Computer Science - log₂ and Bit Counts
Base-2 logarithms are fundamental to computer science and algorithm analysis. Key applications:
- Bits needed to represent n values: ⌈log₂(n)⌉ bits. To represent 256 values: log₂(256) = 8 bits. This is why 8-bit bytes hold values 0–255.
- Binary search complexity: A sorted list of n items takes at most ⌈log₂(n)⌉ comparisons. For 1 million items: log₂(1,000,000) ≈ 20 comparisons maximum.
- Tree depth: A balanced binary tree with n nodes has depth ⌈log₂(n)⌉. This is why log₂ is everywhere in data structure analysis.
- Information theory: A message with probability p contains −log₂(p) bits of information (Shannon entropy).
How this calculator works, and where the numbers come from
The Logarithm Calculator applies the standard formula for this calculation to the values you enter and updates the result as you type. The calculation itself happens in your browser, and the page explains the method so you can check any result by hand.
Please note: Results are provided for general information and are calculated from the values you enter.