Log and ln are both logarithms — the only difference is the base. A plain log means base 10 (the common logarithm), while ln means base e, about 2.71828 (the natural logarithm). So log(1000) = 3 because 10³ = 1000, but ln(1000) ≈ 6.9078 because e^6.9078 = 1000. Same question (“what power gives me 1000?”), different base, different answer. This guide compares them side by side, shows when to use each, and clears up the confusion that costs students exam marks.
What Is the Difference Between Log and Ln?
It is exactly one thing: the base. Every logarithm answers “to what power must I raise the base to get this number?” — log_b(x) = y means b^y = x. The notation just fixes which base you mean:
- log(x) = log₁₀(x): the power you put on 10 to get x. log(1000) = 3.
- ln(x) = log_e(x): the power you put on e (≈ 2.71828) to get x. ln(1000) ≈ 6.9078.
Because 10 and e are different numbers, the same input gives different outputs. There is no deeper mystery — but since textbooks and calculators switch between them without warning, knowing which one you are looking at matters enormously. Calcrio’s free logarithm calculator evaluates both, so you can check any value instantly.
Log vs Ln: Comparison Table
| log (common logarithm) | ln (natural logarithm) | |
|---|---|---|
| Base | 10 | e ≈ 2.71828 |
| Written as | log(x) | ln(x) |
| log(1000) / ln(1000) | 3 (10³ = 1000) | ≈ 6.9078 (e^6.9078 = 1000) |
| Value at its own base | log(10) = 1 | ln(e) = 1 |
| Value at 1 | log(1) = 0 | ln(1) = 0 |
| Typical use | pH, decibels, Richter scale, engineering | calculus, growth/decay, compound interest |
| Change-of-base link | log(x) = ln(x) / ln(10) | ln(x) = log(x) × ln(10) ≈ 2.3026 · log(x) |
The last row is the key relationship: ln(x) = 2.302585 × log(x), because ln(10) ≈ 2.302585. Check it with x = 1000: 2.302585 × 3 = 6.907755 ≈ 6.9078 ✓. The two logs are always proportional — they just measure the same “power” in different units, like inches versus centimetres.
Worked Examples: Same Numbers, Both Logs
Take x = 100 and ask each log for the power:
- log(100) = 2, because 10² = 100.
- ln(100) ≈ 4.6052, because e^4.6052 = 100.
Or x = 20:
- log(20) ≈ 1.3010, because 10^1.3010 ≈ 20.
- ln(20) ≈ 2.9960, because e^2.9960 ≈ 20.
And the cross-check ratio holds in every case: 2.302585 × 1.3010 ≈ 2.9960 ✓. If your two answers don’t stand in this ratio, one of them is wrong — a handy self-check on exams.
When Do You Use Log, and When Ln?
Use log (base 10) when the problem involves powers of 10 or human-made scales: pH (which is −log₁₀ of hydrogen-ion concentration), the Richter earthquake scale, decibels for sound, and most engineering formulas. These scales were designed around base 10 because it makes orders of magnitude easy to read — each whole step is “ten times bigger.”
Use ln (base e) in calculus, physics, and finance: derivatives of exponentials simplify beautifully with e (the derivative of e^x is e^x), so growth and decay, continuous compound interest, half-life, and differential equations all live in ln-land. Whenever you see e^x or a rate “proportional to the current amount,” reach for ln.
The safe rule for exams: if the question says “log” with no base, it means base 10. If it says “ln,” it means base e. When in doubt, the change-of-base formula converts between them: log(x) = ln(x)/ln(10) ≈ ln(x)/2.3026.
Why Does e Get Its Own Logarithm?
Because e ≈ 2.71828 is the natural base of growth: e^x is the unique function that is its own derivative, which makes calculus with e effortless. Since logarithms are the inverse of exponentials, the logarithm that inverts e^x — that is, ln — inherits that elegance: the derivative of ln(x) is simply 1/x. That is the entire reason mathematics gave e’s log its own symbol. Base 10’s log gets the bare “log” name for the opposite reason: it is the one humans read most easily on everyday scales.
How Do You Convert Between Log and Ln?
With the change-of-base formula. To get ln from log: ln(x) = 2.302585 × log(x). To get log from ln: log(x) = ln(x) / 2.302585. Example: ln(1000) ≈ 6.9078, so log(1000) = 6.9078 / 2.302585 = 3 exactly ✓. Any calculator with one of the two buttons can reach the other — and Calcrio’s logarithm calculator plus the exponent calculator let you verify both directions.
Quick Check: Spotting a Mixed-Up Base in 5 Seconds
On an exam, the fastest way to catch a log/ln mix-up is the 2.3 ratio: your ln answer should always be about 2.3 times your log answer. If you computed log(50) ≈ 1.6990, then ln(50) must be ≈ 2.302585 × 1.6990 ≈ 3.9120 — and indeed ln(50) ≈ 3.9120 ✓. If instead you got 1.6990 from the LN button, you pressed the wrong key.
Another sanity check: log of a number between 1 and 10 is always between 0 and 1 (e.g. log(2) ≈ 0.3010), while ln of a number between 1 and e (≈ 2.718) is between 0 and 1 (e.g. ln(2) ≈ 0.6931). Because e < 10, ln answers run larger than log answers for the same input above 1 — if your “ln” came out smaller than your “log,” something is backwards.
And one more trap to file away: log₂ is neither log nor ln. The notation log₂(32) = 5 (since 2⁵ = 32) uses an explicit base subscript, so it sidesteps the whole log-vs-ln ambiguity — always write the base when it isn’t 10 or e.
FAQ
What is the difference between log and ln?
Only the base: log means base 10 and ln means base e (≈ 2.71828). log(1000) = 3 because 10³ = 1000, while ln(1000) ≈ 6.9078 because e^6.9078 = 1000. They are related by ln(x) = 2.302585 × log(x).
Is log always base 10?
In most textbooks, calculators, and engineering contexts, yes — a bare “log” means base 10. In some advanced math contexts authors use “log” for the natural logarithm, but they usually say so explicitly. If your class hasn’t said otherwise, log = log₁₀.
Why is ln called the “natural” logarithm?
Because base e arises naturally in calculus: e^x is its own derivative, and ln(x) differentiates to the simple 1/x. It is the logarithm that makes calculus formulas cleanest.
How do you change ln to log?
Divide by ln(10) ≈ 2.302585: log(x) = ln(x) / 2.302585. For example, ln(20) ≈ 2.9960 gives log(20) ≈ 2.9960 / 2.302585 ≈ 1.3010, and indeed 10^1.3010 ≈ 20 ✓.
Which one do I use on a calculator?
Most calculators have both a LOG button (base 10) and an LN button (base e). Match the button to your problem’s base — mixing them up is one of the most common log errors, since ln(10) ≈ 2.3026 but log(10) = 1.
Can you take log or ln of 0 or a negative number?
No — both are undefined for zero and negative arguments, since no exponent on a positive base produces 0 or a negative number. If a problem seems to ask for one, recheck your algebra; you may have an extraneous solution to reject.