Is log the same as ln? Log vs ln explained

Last reviewed on 2 October 2026.

ln(x)  = 2.302585 × log(x)
log(x) = 0.434294 × ln(x)

Two logarithm keys sit next to each other on the SCI panel of the calculator: log and ln. They look almost identical and they behave almost identically, which is exactly why people pick the wrong one. This page explains what each key computes, how the two are related, which one your textbook or software means, and when each is the natural choice.

Log vs ln at a glance

What a logarithm answers

A logarithm answers a question about an exponent: "what power do I raise the base to in order to get this number?". log₁₀(1000) equals 3 because 10³ is 1000. log₂(8) equals 3 because 2³ is 8. The same idea works for any positive base other than 1.

If exponentiation undoes a logarithm, a logarithm undoes exponentiation. They are inverse operations, and most of the work you do with logarithms is some flavour of "solve for the exponent". log and ln are simply the two bases common enough to get their own keys.

How log and ln are related

Because every logarithm is a scaled copy of every other, ln and log are always in the same ratio: ln(x) is ln(10) ≈ 2.302585 times log(x), for every positive x. Some values side by side:

On a graph the two curves have the same shape. Both cross zero at x = 1, both are negative between 0 and 1, both grow without limit but ever more slowly, and neither is defined for zero or negative numbers. The ln curve is simply stretched vertically by a factor of 2.303.

That also answers a common side question: is ln(x) bigger than log(x)? For x greater than 1, yes — about 2.3 times bigger. For x between 0 and 1, both are negative and ln(x) is the more negative one. At x = 1 they are equal, both zero.

Which "log" does your book, field or software mean?

The keys on a calculator are unambiguous. The word "log" in a formula is not. This is the source of most "is log the same as ln" confusion:

Notation also varies by country. Textbooks in much of continental Europe, Russia and parts of Asia write lg for base 10 and keep log for a general base or for ln; English-language school books more often write log for base 10. If a formula comes from a source you do not know, a quick test is to check a known value: if the source treats log(100) as 2, it means base 10.

When base 10 is the natural choice

Base-10 logarithms shine when the underlying scale is decimal. A few cases where you almost always want log:

• pH. The pH of a solution is −log₁₀ of the hydrogen ion activity. A pH of 4 means an activity of 10⁻⁴ moles per litre.
• Decibels. Sound intensity in decibels is 10 × log₁₀(I / I₀). A 10 dB jump means ten times the intensity.
• Earthquake magnitude. Each whole-number step on the moment magnitude scale corresponds to about 10 times the ground motion and roughly 31.6 times the energy.
• Counting digits. The number of digits in a positive integer N is floor(log₁₀(N)) + 1.
• Log-scale charts. Axes labelled 1, 10, 100, 1000 are base-10 logarithmic scales.

When natural log is the natural choice

Natural logarithms come up wherever continuous growth or continuous decay is involved, because the exponential function eˣ has the special property that its rate of change is itself.

• Compound interest, continuous case. Money growing at rate r compounded continuously turns into P × e^(rt) after t years. To solve for time given a final amount, take ln of both sides.
• Half-life and decay. A quantity that decays exponentially follows N(t) = N₀ × e^(−kt). The half-life is ln(2) / k, roughly 0.693 over the decay constant.
• Calculus. The derivative of ln(x) is 1/x and the integral of 1/x is ln|x| + C. The derivative of log₁₀(x) is 1 / (x × ln(10)) — clean only with the natural log. This is why calculus books write "log" and mean ln.
• Statistics and economics. Log returns, log-linear regression, elasticities and log-likelihoods all use ln. A useful property: for small changes, the difference in ln is approximately the percentage change. ln(1.05) ≈ 0.0488, close to 5%; log(1.05) ≈ 0.0212 has no such reading.
• Machine learning and AI. Cross-entropy loss, log-probabilities and softmax are defined with natural logs. When a paper writes log without a base, assume ln.
• Information theory. Entropy measured with ln is in nats; with log₂ it is in bits. The factor between the two is ln(2).

Is log base 2 the same as ln?

No. log₂ answers "2 to what power gives x?", ln answers "e to what power gives x?". Because 2 is smaller than e, log₂ grows faster: log₂(x) = ln(x) / ln(2) ≈ 1.4427 × ln(x). For example, log₂(8) = 3 but ln(8) ≈ 2.079. The calculator has no log₂ key; type ln(8) / ln(2) or log(8) / log(2) instead.

Solving exponential equations: does it matter which key you use?

Not for the final answer, as long as you use the same key on both sides. To solve 2ˣ = 50, take a logarithm of both sides and divide:

x = log(50) / log(2)  = 5.643856…
x = ln(50)  / ln(2)   = 5.643856…

Both give the same result because the conversion factor cancels. The choice only matters when the base of the exponential is e — then ln undoes it in one step (ln(eˣ) = x) and log would leave a stray factor of 0.4343. The laws of logarithms — product, quotient and power rules — are identical for log, ln and every other base.

The change-of-base formula

If you only have log and ln on the calculator but you need a logarithm in some other base — say base 2 or base 7 — use the change-of-base identity:

log_b(x) = log(x) / log(b)
log_b(x) = ln(x) / ln(b)

Both forms are correct; pick whichever key is closer. To compute log₂(64), type log(64) / log(2) and the calculator returns 6. To compute log₇(343), type ln(343) / ln(7) and you get 3. The same trick lets you reproduce one logarithm key from the other: log(x) = ln(x) / ln(10).

Worked example: continuously compounded interest

Suppose a deposit of 1,000 grows to 1,500 at a continuously compounded interest rate of 4% per year. How long does that take? The model is 1500 = 1000 × e^(0.04 × t). Divide both sides by 1,000, then take ln:

ln(1.5) = 0.04 × t
t = ln(1.5) / 0.04

Type ln(1.5) / 0.04 into the calculator and the result is about 10.14 years. Trying to solve the same problem with log works only if you also divide by log(e), which is roughly 0.4343 — easy to forget. This is the kind of problem where ln is unambiguously the right key.

Worked example: pH

A solution has a hydrogen ion activity of 4 × 10⁻⁹ mol/L. The pH is −log(4 × 10⁻⁹). Type -log(4 * 10^-9) and you get about 8.40. Using ln instead would give about 19.3 — wrong by a factor of 2.303, because the chemistry definition is locked to base 10.

Worked example: digits in a big number

How many digits does 2¹⁰⁰ have? Type log(2^100) and you get about 30.103. The integer part plus one is the digit count: 31 digits. The same calculation with ln would give a different number that needs a conversion factor, so log is the cleaner choice here.

Common mistakes

• Mixing up the keys mid-problem. If a textbook formula looks like −log(...) for pH or decibels, never substitute ln. They differ by a factor of about 2.303.
• Copying a formula from code or a paper into a calculator. np.log(x) or a "log" in a statistics paper is ln. Press ln, not log.
• Taking the log of zero or a negative number. Both keys return an error. If your problem is heading there, double-check the model — the input to a real logarithm must be strictly positive.
• Forgetting parentheses. log10x looks fine on paper but on the calculator you must type log(10x) or log(10) × x, depending on what you mean. The keys insert log( or ln( for you, which helps.
• Assuming log(a + b) = log(a) + log(b). The product rule is log(a × b) = log(a) + log(b). There is no simple rule for the log of a sum, in any base.

Decision checklist

• Is the formula about continuous growth, decay, calculus, statistics or machine learning? Use ln.
• Is the formula about pH, decibels, magnitudes, or digit counting? Use log.
• Solving aˣ = b for x? Either key works: log(b) / log(a).
• Do you need a different base entirely? Apply the change-of-base formula.
• Is the input zero or negative? Stop and re-check the problem.

Frequently asked questions