Arithmetic
Logarithm Calculator
Compute a logarithm in any base — common (log₁₀), natural (ln), base 2, or a base you choose — with the change-of-base working, the antilog check, and a full lookup chart. Results update as you type.
Compute a logarithm
base 10Base
log₁₀(1000) =3
Exact — 10 raised to 3 is exactly 1000
Inverse check (antilog)
103 = 1000
log₁₀ x
3
ln x
6.9078
log₂ x
9.9658
Try an example
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The basics
What is a logarithm?
A logarithm is the inverse of raising to a power. It answers a single question: to what exponent must a base be raised to land on a given number? Where an exponent packs repeated multiplication, a logarithm unpacks it — which is why logs turn multiplying into adding and shrink runaway ranges down to a readable scale.
logₐ(x) = y ⇔ aʸ = x
It undoes an exponent
The log and the power are two readings of the same fact. Given a base a and a number x, the log returns the exponent y that gets you there.
log₁₀(1000) = 3 because 10³ = 1000
log = base 10 · ln = base e
The two everyday bases
Written bare, log usually means base ten, the common log. The natural log ln uses base e ≈ 2.718, the base that appears naturally in growth and calculus.
log(100) = 2 · ln(e) = 1
Method
How to find a logarithm by hand
Three steps work for any base. The only real machinery is the change-of-base formula, and even that is just one division.
- 1
Rewrite it as an exponent question
Turn logₐ(x) into “the base a raised to what power gives x?” This reframing alone solves every exact case.
log₂(32) → 2^? = 32 - 2
Spot the exponent, or change the base
If x is a clean power of the base, read the exponent straight off. Otherwise rewrite using natural logs, which every calculator can evaluate.
2^5 = 32 → 5 or ln(x) ÷ ln(a) - 3
Evaluate and check by exponentiating
Compute the division to get the logarithm, then raise the base to that answer to confirm you land back on x.
ln(32) ÷ ln(2) = 5 → 2^5 = 32 ✓
Worked examples
See it across bases
The same procedure runs whether the base is ten, e, two, or something custom. Switch tabs to watch each one compute.
Find log₁₀(1000)
As an exponent: 10? = 1000
Base 10 is built in, so read it directly.
Evaluate: 103 = 1000 (exact)
log₁₀(1000) = 3
Where it shows up
Why logarithms matter
Logs are the quiet machinery behind any scale that has to span many orders of magnitude.
pH = −log₁₀[H⁺]
Chemistry & sound
The pH scale and the decibel are both logarithmic, so a step of one means a tenfold change in concentration or power.
O(log n)
Computer science
Binary search and balanced trees run in logarithmic time — doubling the data adds only one extra step. Base 2 logs also count bits of information.
M = log₁₀(A) + …
Earth & finance
The Richter scale measures quakes, and log returns linearize compounding growth, so a straight line on a log chart means a steady percentage gain.
The laws
Logarithm rules worth remembering
Every log law follows from one idea — logs undo exponents — so each rule mirrors an exponent rule. These eight cover almost everything you will meet.
logₐ(xy) = logₐ x + logₐ y
Product rule
The log of a product is the sum of the logs. Multiplication turns into addition.
log₁₀(2·5) = log₁₀ 2 + log₁₀ 5 = 1
logₐ(x/y) = logₐ x − logₐ y
Quotient rule
The log of a quotient is the difference of the logs. Division turns into subtraction.
log₁₀(100/10) = 2 − 1 = 1
logₐ(xⁿ) = n · logₐ x
Power rule
An exponent inside a log comes out front as a multiplier — how logs tame big powers.
log₁₀(10³) = 3 · log₁₀ 10 = 3
logₐ x = ln x ÷ ln a
Change of base
Rewrite any base using natural or common logs — the trick your calculator runs internally.
log₂ 8 = ln 8 ÷ ln 2 = 3
logₐ a = 1
Log of the base
Any base raised to the first power is itself, so its own log is always one.
log₇ 7 = 1, ln e = 1
logₐ 1 = 0
Log of one
Any nonzero base raised to the zero power is one, so the log of one is always zero.
log₁₀ 1 = 0, ln 1 = 0
logₐ(1/x) = −logₐ x
Reciprocal rule
Taking a reciprocal flips the sign of the log — a special case of the quotient rule.
log₁₀(1/100) = −2
a^(logₐ x) = x
Inverse with exponent
Exponentiation and logarithm undo each other, since they are inverse operations.
10^(log₁₀ 5) = 5
Reference
Logs of 1 through 10, every base
The classic log-table row: the common, natural, and binary log of each digit, rounded to four places.
| x | log₁₀ x | ln x | log₂ x |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 0.301 | 0.6931 | 1 |
| 3 | 0.4771 | 1.0986 | 1.585 |
| 4 | 0.6021 | 1.3863 | 2 |
| 5 | 0.699 | 1.6094 | 2.3219 |
| 6 | 0.7782 | 1.7918 | 2.585 |
| 7 | 0.8451 | 1.9459 | 2.8074 |
| 8 | 0.9031 | 2.0794 | 3 |
| 9 | 0.9542 | 2.1972 | 3.1699 |
| 10 | 1 | 2.3026 | 3.3219 |
Powers of ten
The cleanest common logs
For powers of ten the common log is just the exponent — the reason log₁₀ is the natural fit for counting digits and orders of magnitude.
log₁₀ 0.001
-3
log₁₀ 0.01
-2
log₁₀ 0.1
-1
log₁₀ 1
0
log₁₀ 10
1
log₁₀ 100
2
log₁₀ 1,000
3
log₁₀ 10,000
4
log₁₀ 100,000
5
log₁₀ 1,000,000
6
Quick chart
The log of any number, 1 to 100
Switch the base and scan the value you need — common, natural, or binary log of every integer from one to a hundred, no typing required.
log₁₀(n) for n = 1 to 100
log₁₀ 1
0
log₁₀ 2
0.301
log₁₀ 3
0.4771
log₁₀ 4
0.6021
log₁₀ 5
0.699
log₁₀ 6
0.7782
log₁₀ 7
0.8451
log₁₀ 8
0.9031
log₁₀ 9
0.9542
log₁₀ 10
1
log₁₀ 11
1.0414
log₁₀ 12
1.0792
log₁₀ 13
1.1139
log₁₀ 14
1.1461
log₁₀ 15
1.1761
log₁₀ 16
1.2041
log₁₀ 17
1.2304
log₁₀ 18
1.2553
log₁₀ 19
1.2788
log₁₀ 20
1.301
log₁₀ 21
1.3222
log₁₀ 22
1.3424
log₁₀ 23
1.3617
log₁₀ 24
1.3802
log₁₀ 25
1.3979
log₁₀ 26
1.415
log₁₀ 27
1.4314
log₁₀ 28
1.4472
log₁₀ 29
1.4624
log₁₀ 30
1.4771
log₁₀ 31
1.4914
log₁₀ 32
1.5051
log₁₀ 33
1.5185
log₁₀ 34
1.5315
log₁₀ 35
1.5441
log₁₀ 36
1.5563
log₁₀ 37
1.5682
log₁₀ 38
1.5798
log₁₀ 39
1.5911
log₁₀ 40
1.6021
log₁₀ 41
1.6128
log₁₀ 42
1.6232
log₁₀ 43
1.6335
log₁₀ 44
1.6435
log₁₀ 45
1.6532
log₁₀ 46
1.6628
log₁₀ 47
1.6721
log₁₀ 48
1.6812
log₁₀ 49
1.6902
log₁₀ 50
1.699
log₁₀ 51
1.7076
log₁₀ 52
1.716
log₁₀ 53
1.7243
log₁₀ 54
1.7324
log₁₀ 55
1.7404
log₁₀ 56
1.7482
log₁₀ 57
1.7559
log₁₀ 58
1.7634
log₁₀ 59
1.7709
log₁₀ 60
1.7782
log₁₀ 61
1.7853
log₁₀ 62
1.7924
log₁₀ 63
1.7993
log₁₀ 64
1.8062
log₁₀ 65
1.8129
log₁₀ 66
1.8195
log₁₀ 67
1.8261
log₁₀ 68
1.8325
log₁₀ 69
1.8388
log₁₀ 70
1.8451
log₁₀ 71
1.8513
log₁₀ 72
1.8573
log₁₀ 73
1.8633
log₁₀ 74
1.8692
log₁₀ 75
1.8751
log₁₀ 76
1.8808
log₁₀ 77
1.8865
log₁₀ 78
1.8921
log₁₀ 79
1.8976
log₁₀ 80
1.9031
log₁₀ 81
1.9085
log₁₀ 82
1.9138
log₁₀ 83
1.9191
log₁₀ 84
1.9243
log₁₀ 85
1.9294
log₁₀ 86
1.9345
log₁₀ 87
1.9395
log₁₀ 88
1.9445
log₁₀ 89
1.9494
log₁₀ 90
1.9542
log₁₀ 91
1.959
log₁₀ 92
1.9638
log₁₀ 93
1.9685
log₁₀ 94
1.9731
log₁₀ 95
1.9777
log₁₀ 96
1.9823
log₁₀ 97
1.9868
log₁₀ 98
1.9912
log₁₀ 99
1.9956
log₁₀ 100
2
Highlighted values are exact whole-number logs (powers of the base).
Watch out
Common logarithm mistakes
Most log errors come from treating a log like ordinary multiplication. These four trip up almost everyone at first.
logₐ(x + y) ≠ logₐ x + logₐ y
The sum trap
There is no rule for the log of a sum. The product rule splits a log of a product, never a log of a sum — those stay locked together.
logₐ(x) ÷ logₐ(y) ≠ logₐ(x − y)
Division is not subtraction here
Dividing two logs is the change-of-base move, not the quotient rule. The quotient rule applies to the log of a quotient, which is a different thing entirely.
x > 0
No logs of zero or negatives
Over the reals the input must be strictly positive. A negative or zero argument has no real log, so a calculator returns an error.
(logₐ x)ⁿ ≠ n · logₐ x
Where the exponent lives
The power rule moves an exponent that sits on the input. An exponent on the whole log stays put — squaring a log is not the same as logging a square.
Glossary
Logarithm terms, defined
The handful of words that show up around logarithms.
Base
The fixed number being raised to a power, the a in logₐ. Common choices are 10, e, and 2.
Argument
The number you take the log of, the x inside the parentheses. It must be positive.
Mantissa & characteristic
For a common log, the whole-number part is the characteristic and the decimal part is the mantissa — the split that made paper log tables work.
Antilogarithm
The inverse operation: raising the base to the log to recover the original number. The tool shows this as its check step.
FAQ
Common questions
A logarithm answers the question "to what power must a base be raised to produce a given number?" If logₐ(x) = y, then a raised to the power y equals x. For example, log₁₀(1000) = 3 because 10³ = 1000.
"log" written on its own usually means the common logarithm, base 10. "ln" is the natural logarithm, base e (about 2.718281828). So log(100) = 2 because 10² = 100, while ln(e) = 1 because e¹ = e. On many calculators "log" is base 10 and "ln" is base e.
The change of base formula lets you compute a logarithm in any base from natural or common logs: logₐ(x) = ln(x) ÷ ln(a). For example log₂(8) = ln(8) ÷ ln(2) = 3. This is how a calculator evaluates bases it has no dedicated button for.
Not over the real numbers. The logarithm is only defined for values greater than zero, because no real power of a positive base produces a negative number or zero. The log of a negative number exists only in the complex numbers, and log(0) tends to negative infinity.
The logarithm of 1 is always 0, in every base, because any nonzero base raised to the power 0 equals 1. So log₁₀(1) = 0, ln(1) = 0, and log₂(1) = 0.
For exact powers, count them: log₁₀(1000) = 3 because 1000 is 10³, and log₂(32) = 5 because 32 is 2⁵. For other values, use the log laws to break the number into factors you know — log₁₀(200) = log₁₀(2) + log₁₀(100) ≈ 0.301 + 2 = 2.301 — or read a value from a log table.
The antilogarithm is the inverse of a logarithm: it raises the base back to a power. If log₁₀(x) = 2 then the antilog is 10² = 100. In general, the antilog of y in base a is aʸ, which recovers the original number.
Logarithms turn multiplication into addition and huge ranges into manageable scales. They underpin the pH scale in chemistry, decibels in sound, the Richter scale for earthquakes, the runtime of efficient algorithms (O(log n)), and information measured in bits.
Keep going
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Written by
UC
Author
Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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