Logarithm Calculator — log₁₀, ln (Natural Log) & Any Base
Reviewed by CalcMulti Editorial Team·Last updated: February 2026
Calculate log base 10, natural log (ln), log base 2, or any custom base instantly. Enter a number and base — see the result with step-by-step explanation.
Common Log Values
Result
About Logarithms: A logarithm answers the question: "To what power must the base be raised, to produce a given number?" This calculator supports base 10, natural logarithm (base e), and any positive custom base.
Practical Applications: Useful in mathematics, physics, engineering, finance, and data analysis for exponential growth, sound intensity, and pH calculations.
What Are Logarithms?
A logarithm is the inverse operation of exponentiation. It answers the question: "To what power must a given base be raised to produce a specific number?" In mathematical notation, if b^y = x, then the logarithm base b of x equals y.
logb(x) = y means by = x
For example, log10(1000) = 3 because 10 raised to the 3rd power equals 1000. Logarithms are used extensively in science, engineering, and finance to handle quantities that span many orders of magnitude.
Fundamental Logarithm Rules
Three core rules govern how logarithms behave. Mastering these rules makes it possible to simplify and solve complex logarithmic expressions.
Product Rule: logb(x * y) = logb(x) + logb(y)
Quotient Rule: logb(x / y) = logb(x) - logb(y)
Power Rule: logb(xn) = n * logb(x)
Additionally, the Change of Base Formula lets you convert between bases: logb(x) = logk(x) / logk(b). This is particularly useful when your calculator only supports base-10 or natural logarithms.
Natural Log vs. Common Log
The two most widely used logarithm bases are 10 and e (approximately 2.71828). The common logarithm (log10, often written simply as "log") is used in everyday calculations and in fields like chemistry (pH scale) and acoustics (decibels). The natural logarithm (loge, written as "ln") is central to calculus, continuous growth models, and most branches of higher mathematics.
Common Logarithm Values
| x | log10(x) | ln(x) |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 0.3010 | 0.6931 |
| e (2.718) | 0.4343 | 1 |
| 10 | 1 | 2.3026 |
| 100 | 2 | 4.6052 |
| 1000 | 3 | 6.9078 |
Real-World Applications of Logarithms
Logarithms are far from purely theoretical -- they appear in many practical contexts:
- Richter Scale (Earthquakes): The Richter scale is logarithmic. An earthquake measuring 6.0 is ten times more powerful than one measuring 5.0, meaning each whole-number increase represents a tenfold increase in measured amplitude.
- Decibels (Sound Intensity): Sound levels are measured in decibels using a logarithmic scale. A sound at 80 dB is ten times more intense than one at 70 dB. This allows a vast range of intensities to be expressed in manageable numbers.
- pH Scale (Chemistry): The pH of a solution is defined as pH = -log10[H+], where [H+] is the hydrogen ion concentration. A pH change of 1 unit represents a tenfold change in acidity.
- Compound Interest (Finance): The natural logarithm is used to calculate the time required for an investment to reach a target value under continuous compounding.
How to Calculate Logarithms — Step by Step
Four methods for four common scenarios. Choose the one that matches your problem.
Method 1 — Base 10 (common log): log₁₀(500)
- Step 1: Express as a product: 500 = 5 × 100 = 5 × 10²
- Step 2: Apply product rule: log(500) = log(5) + log(100)
- Step 3: Use known values: 0.69897 + 2 = 2.69897
Method 2 — Natural log (ln): ln(20)
- Step 1: Break down: 20 = 4 × 5 = 2² × 5
- Step 2: Apply rules: ln(20) = 2·ln(2) + ln(5)
- Step 3: Substitute: 2(0.6931) + 1.6094 = 1.3863 + 1.6094 = 2.9957
Method 3 — Custom base (log₅ of 125)
- Step 1: Try integer powers: 5¹=5, 5²=25, 5³=125 ✓
- Answer directly: log₅(125) = 3
- Alternative (change of base): log(125)/log(5) = 2.0969/0.6990 = 3
Method 4 — Non-integer result (log₂(10))
- Step 1: No integer power works → use change of base formula
- Step 2: log₂(10) = log(10) / log(2) = 1 / 0.30103
- Step 3: = 3.32193
This means 10 in binary requires about 3.32 bits — the origin of the approximation "10 bits ≈ 3 decimal digits."
Step-by-Step Example: log₂(64)
Let's calculate log₂(64) — the power to which 2 must be raised to equal 64.
Step 1: Identify base = 2, number = 64
Step 2: Apply change of base formula: log₂(64) = ln(64) / ln(2)
Step 3: ln(64) = 4.1589, ln(2) = 0.6931
Step 4: 4.1589 / 0.6931 = 6
Verify: 2⁶ = 64 ✓
The answer is 6, because 2 must be raised to the 6th power to produce 64. You can verify any logarithm result by checking that base^result equals the original number.
Common Mistakes with Logarithms
Mistake 1: log(a + b) ≠ log(a) + log(b)
The product rule applies to multiplication, not addition. log(a + b) cannot be simplified further. Only log(a × b) = log(a) + log(b).
Mistake 2: Confusing log₁₀ and ln
In most textbooks, "log" without a base means log₁₀. In many programming languages and scientific contexts, "log" means the natural log. Always clarify which base is intended to avoid errors.
Mistake 3: Taking log of a negative number
log(x) is only defined for x > 0 in real numbers. Attempting to compute log(−5) will return an error or undefined. Check your equation setup if this occurs.
Mistake 4: log(x/y) = log(x) / log(y) — Wrong!
The quotient rule is log(x/y) = log(x) − log(y), not division of logs. log(x) / log(y) is actually the change of base formula for log_y(x).
Logarithm Quick Reference Table
Memorize these common values — they appear frequently in exams, engineering, and science.
| Expression | Value | Explanation |
|---|---|---|
| log₁₀(1) | 0 | 10⁰ = 1 |
| log₁₀(10) | 1 | 10¹ = 10 |
| log₁₀(100) | 2 | 10² = 100 |
| log₁₀(1000) | 3 | 10³ = 1,000 |
| ln(1) | 0 | e⁰ = 1 |
| ln(e) | 1 | e¹ = e |
| ln(2) | ≈ 0.693 | Doubling time rule: 0.693/r |
| log₂(8) | 3 | 2³ = 8 (3 bits = 8 values) |
| log₂(1024) | 10 | 2¹⁰ = 1,024 (1 KB = 2¹⁰ bytes) |
Related Math Tools
Logarithms are closely related to exponents and appear throughout statistics and algebra.
Antilogarithm: Reversing the Logarithm
The antilogarithm (antilog) is the inverse operation of the logarithm. If logb(x) = y, then antilogb(y) = by = x. It answers: "What number has this logarithm?"
Base-10 antilog
antilog₁₀(y) = 10ʸ
antilog₁₀(3) = 10³ = 1,000
Natural antilog
antiln(y) = eʸ
antiln(2) = e² ≈ 7.389
| Logarithm | Value (y) | Antilog = b^y |
|---|---|---|
| log₁₀(x) = 2 | 2 | 10² = 100 |
| log₁₀(x) = 3.5 | 3.5 | 10^3.5 ≈ 3,162 |
| ln(x) = 1 | 1 | e¹ ≈ 2.718 |
| ln(x) = 4 | 4 | e⁴ ≈ 54.60 |
Doubling Time and Half-Life Using ln
Natural logarithm is the key to calculating how long exponential processes take to double or halve. Two of the most useful formulas in science and finance:
Doubling Time
T₂ = ln(2) / r ≈ 0.693 / r
Also expressed as the Rule of 72: T₂ ≈ 72 / (r × 100)
Half-Life
T½ = ln(2) / λ ≈ 0.693 / λ
Used for radioactive decay, drug clearance, and exponential decay processes.
| Scenario | Rate (r) | Doubling Time | Formula |
|---|---|---|---|
| S&P 500 (avg ~10%) | 10%/yr | ~7.3 years | ln(2)/0.10 |
| Savings account (5%) | 5%/yr | ~13.9 years | ln(2)/0.05 |
| E. coli doubling | ~100%/20 min | 20 minutes | ln(2)/ln(2) × 20 min |
| Carbon-14 decay | decay constant λ | 5,730 years | ln(2)/λ |
Logarithmic Scales: Richter, Decibels, and pH
Many measurement systems use logarithmic scales because the phenomena they measure span many orders of magnitude. A linear scale would require enormous numbers to represent the full range.
| Scale | Formula | +1 unit means | Example |
|---|---|---|---|
| Richter (earthquakes) | M = log₁₀(A/A₀) | 10× more wave amplitude | M7 is 10× stronger than M6 |
| Decibels (sound) | dB = 10·log₁₀(I/I₀) | ~26% more intensity | Jet engine: 140 dB vs library: 40 dB (10¹⁰ more intense) |
| pH (acidity) | pH = −log₁₀[H⁺] | 10× more acidic | Lemon (pH 2) is 100× more acidic than coffee (pH 4) |
pH Worked Example
If [H⁺] = 0.001 mol/L (10⁻³), then pH = −log₁₀(10⁻³) = −(−3) = 3. This corresponds to vinegar — strongly acidic. At pH 7 (neutral water), [H⁺] = 10⁻⁷ mol/L. The 4-unit difference means vinegar has 10⁴ = 10,000 times more hydrogen ions than water.
Binary Logarithm (log₂) in Computer Science
Base-2 logarithm is fundamental in computer science and algorithm analysis. It answers: "How many bits are needed?" and "How many times can you halve this?"
Algorithm Complexity (Big O)
Binary search takes log₂(n) steps to find an element in a sorted list of n items. For a list of 1,024 items: log₂(1024) = 10 comparisons maximum. For 1,000,000 items: log₂(10⁶) ≈ 20 comparisons.
Bit Count
To store n different values you need ⌈log₂(n)⌉ bits. For 256 values: log₂(256) = 8 bits (1 byte). For 65,536 values: log₂(65,536) = 16 bits (2 bytes). For 1,000 values: ⌈log₂(1000)⌉ = ⌈9.97⌉ = 10 bits.
| n values | log₂(n) bits | Binary search steps | Use case |
|---|---|---|---|
| 2 | 1 bit | 1 | Boolean (true/false) |
| 256 | 8 bits | 8 | ASCII characters (1 byte) |
| 65,536 | 16 bits | 16 | Basic Unicode (2 bytes) |
| 1,000,000 | ≈20 bits | 20 | Large database lookup |
Logarithm vs Exponent: Inverse Operations
Logarithms and exponents are inverse operations — each undoes the other. Understanding this relationship is the key to solving exponential equations.
| Question | Use exponent | Use logarithm |
|---|---|---|
| What is 2 to the power 10? | 2¹⁰ = 1,024 | — |
| How many doublings to reach 1,024? | — | log₂(1024) = 10 |
| $1,000 growing at 7% for 20 years? | 1000×1.07²⁰ = $3,870 | — |
| How long to reach $3,870 at 7%? | — | ln(3.87)/ln(1.07) ≈ 20 yrs |
Common Logarithm Values — Quick Reference Table
log₁₀ (common log), ln (natural log, base e ≈ 2.71828), and log₂ (binary log) for frequently used numbers.
| x | log₁₀(x) | ln(x) | log₂(x) | Note |
|---|---|---|---|---|
| 0.1 | −1 | −2.3026 | −3.3219 | 10⁻¹ |
| 0.5 | −0.3010 | −0.6931 | −1 | ½ |
| 1 | 0 | 0 | 0 | Always 0 |
| 2 | 0.3010 | 0.6931 | 1 | Key value |
| e ≈ 2.718 | 0.4343 | 1 | 1.4427 | ln(e) = 1 |
| 3 | 0.4771 | 1.0986 | 1.5850 | |
| 4 | 0.6021 | 1.3863 | 2 | 2² |
| 5 | 0.6990 | 1.6094 | 2.3219 | |
| 7 | 0.8451 | 1.9459 | 2.8074 | |
| 8 | 0.9031 | 2.0794 | 3 | 2³ |
| 10 | 1 | 2.3026 | 3.3219 | log₁₀=1 always |
| 16 | 1.2041 | 2.7726 | 4 | 2⁴ |
| 20 | 1.3010 | 2.9957 | 4.3219 | |
| 25 | 1.3979 | 3.2189 | 4.6439 | 5² |
| 50 | 1.6990 | 3.9120 | 5.6439 | |
| 100 | 2 | 4.6052 | 6.6439 | 10² |
| 1,000 | 3 | 6.9078 | 9.9658 | 10³ |
| 10,000 | 4 | 9.2103 | 13.2877 | 10⁴ |
| 1,000,000 | 6 | 13.8155 | 19.9316 | 10⁶ |
Antilogarithm Quick Reference
Common Logarithm Examples — Step-by-Step
Instant answers and explanations for the most commonly searched logarithm calculations. Each example shows the method so you can apply it to any number.
What is log(2)?
log₁₀(2) = 0.30103 — this means 10 raised to the power 0.30103 equals 2.
Verify: 10^0.30103 ≈ 2 ✓
ln(2) = 0.6931 (natural log)
Why it matters: log(2) ≈ 0.301 is one of the most useful values to memorize. It powers the Rule of 72 (doubling time ≈ 72 / interest rate %) and explains why base-10 numbers gain a digit every time they pass a power of 10.
What is log(3)?
log₁₀(3) = 0.47712
ln(3) = 1.0986
log₂(3) = 1.5850
Useful fact: log(3) + log(4) = log(12) using the product rule. Also: log(6) = log(2) + log(3) = 0.301 + 0.477 = 0.778.
What is log(5)?
log₁₀(5) = 0.69897
Shortcut: log(5) = log(10/2) = log(10) − log(2) = 1 − 0.301 = 0.699
This shortcut (log(5) = 1 − log(2)) is a classic exam trick — you only need to memorize log(2).
What is log(50)?
log₁₀(50) = 1.69897
Or: log(100/2) = log(100) − log(2) = 2 − 0.301 = 1.699
What is log(200)?
log₁₀(200) = 2.30103
What is ln(2)?
ln(2) = 0.6931 — this is one of the most important constants in mathematics.
Verify: e^0.6931 ≈ 2 ✓
Why it's important: Doubling time = ln(2) / growth rate. At 7% annual growth, doubling time = 0.693 / 0.07 ≈ 9.9 years. This is the basis of the Rule of 70.
What is ln(10)?
ln(10) = 2.302585…
This is the conversion factor: ln(x) = log₁₀(x) × 2.302585
Conversion formula: To convert any common log to natural log, multiply by ln(10): ln(x) = log₁₀(x) × 2.3026. Example: ln(100) = log(100) × 2.3026 = 2 × 2.3026 = 4.6052.
What is ln(1.038)?
ln(1.038) = 0.037289… — commonly used in continuous compounding and growth-rate calculations.
Approximation: for small x, ln(1+x) ≈ x, so ln(1.038) ≈ 0.038 (within 2%)
Exact: ln(1.038) = 0.037289
Finance use case: If an investment grows at 3.8% per year continuously, the effective annual rate = e^0.038 − 1 ≈ 3.87%. This is also used to convert a stated annual rate to a continuous growth rate: continuous rate = ln(1 + APR).
What is ln(0.05)?
ln(0.05) = −2.9957… — the natural log of values less than 1 is always negative.
= 1.6094 − 4.6052 = −2.9957
Verify: e^(−2.9957) ≈ 0.05 ✓
Why it appears: ln(0.05) = −2.996 is the critical value in p-value calculations (significance threshold α = 0.05) and in half-life problems where a substance decays to 5% of its original amount: t = −ln(0.05) / k = 2.996 / k.
What is log base 2 of 32?
log₂(32) = 5
Step 2: 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32
Answer: log₂(32) = 5
Check: 2⁵ = 32 ✓
Computer science use: A list of 32 items requires at most 5 steps in binary search. 32 different values can be represented with 5 bits.
What is log base 3 of 27?
log₃(27) = 3
Step 2: 3¹ = 3, 3² = 9, 3³ = 27
Answer: log₃(27) = 3
Check: 3³ = 27 ✓
What is log base 4 of 64?
log₄(64) = 3
Alternative: log₄(64) = log(64)/log(4) = 1.806/0.602 = 3
What is log10(0.05)? / What is log(0.05)?
log₁₀(0.05) = −1.3010
= 0.69897 − 2 = −1.30103
Verify: 10^(−1.301) = 0.05 ✓
ln(0.05) = log(0.05) × 2.302585 = −2.9957
Why it appears: log(0.05) = −1.301 shows up in statistics (significance threshold α = 0.05), chemistry (pH calculations), and signal processing (−13 dB attenuation corresponds to a power ratio of 0.05).
What is ln(100,000,000)? / ln(10^8)?
ln(100,000,000) = ln(10⁸) = 8 × ln(10) = 18.4207
log₁₀(100,000,000) = log(10^8) = 8 (exact)
log₂(100,000,000) = log(10^8)/log(2) = 8/0.30103 = 26.575
Pattern: ln(10^n) = n × 2.3026. So ln(1,000) = 6.908, ln(1,000,000) = 13.816, ln(10^8) = 18.421. This is used in information theory, entropy calculations, and Big-O analysis.
What is log10(1.475)?
log₁₀(1.475) = 0.16884
log(1.5) = log(3/2) = log(3) − log(2) = 0.4771 − 0.3010 = 0.1761
Exact: log₁₀(1.475) = 0.16884
ln(1.475) = 0.16884 × 2.302585 = 0.38866
Use case: pH = −log[H⁺]. A hydrogen ion concentration of 1.475 × 10⁻³ mol/L gives pH = 3 − 0.169 = 2.831. Also used in financial calculations: log(1.475) is the log of a 47.5% total return.
What is log(0) or log of a negative number?
log(0) = −∞ (undefined/negative infinity)
log(−x) = undefined in real numbers.
log₁₀(−1) → undefined (no real solution)
log₁₀(−5) → undefined (no real solution)
Logarithms are only defined for positive real numbers. If you get a negative input, check your equation — you may have set up a sign error upstream.
Solving Logarithmic Equations — 3 Templates
Template 1: log_b(x) = n → x = b^n
Example: log₁₀(x) = 3
Solution: x = 10³ = 1,000
Example: log₂(x) = 8
Solution: x = 2⁸ = 256
Template 2: b^x = n → x = log_b(n)
Example: 10^x = 500
Solution: x = log₁₀(500) = 2.699
Example: e^x = 20
Solution: x = ln(20) = 2.996
Template 3: Change of base formula
log_b(x) = log(x) / log(b) = ln(x) / ln(b)
Example: log₅(125) = log(125)/log(5) = 2.097/0.699 = 3
Check: 5³ = 125 ✓