The natural logarithm formula

The natural logarithm, written as ln(x), is a logarithm with base e, where e ≈ 2.718281828…. In plain words: ln(x) tells you what exponent you need to raise e to in order to get x.

The defining relationship is:
y = ln(x) ⇔ e^y = x

This “inverse relationship” is the reason ln is so useful. Exponentials grow (or shrink) very quickly, and logarithms “undo” that growth to reveal the exponent hiding inside. For example: ln(e⁵) = 5 because the ln cancels the e-exponent. Similarly, e^ln(7) = 7.

You can compute logs in any base using ln via the change-of-base formula:
log_b(x) = ln(x) / ln(b)

That’s why this calculator can show log₁₀(x), log₂(x), or log_b(x) after it computes ln(x). The rule works because both sides represent the same exponent (just expressed in different bases).

For real numbers, ln(x) is defined only for x > 0. That’s because the exponential function e^y is always positive for real y, so it can never produce 0 or a negative number. (Complex logarithms exist, but they’re beyond the scope of this page.)

What the calculator does

Under the hood, the calculator uses JavaScript’s Math.log(x) which returns the natural log. Then it optionally converts that value to other bases using change-of-base: log_b(x) = ln(x) / ln(b).

• Validate input: confirm x is a number and x > 0.
• Compute ln(x): ln = Math.log(x).
• Optional conversion: base 10, base 2, or custom base b (b>0 and b≠1).
• Round: to your selected precision.
• Explain: show the relationship e^ln(x)=x and (if selected) the change-of-base step.

Pro tip: If your ln(x) is negative, your x must be between 0 and 1.

Natural log examples (with interpretation)

Seeing examples makes ln feel way less mysterious. In every example below, remember the translation: ln(x) = y means e^y = x.

Enter x=10. The calculator returns ln(10) ≈ 2.3026. Interpretation: e^2.3026 ≈ 10. So 2.3026 is the exponent that turns e into 10.

Enter x=0.5. The result is ln(0.5) ≈ -0.6931. Interpretation: e^-0.6931 ≈ 0.5. Negative ln values happen because you need a negative exponent to shrink e down below 1.

Suppose you want log₁₀(50). Choose “Also show log base 10” and enter x=50. The calculator computes ln(50) and then divides by ln(10). You’ll see log₁₀(50) ≈ 1.6990, meaning 10^1.6990 ≈ 50.

Solve 3·e^2t = 50. Divide both sides by 3: e^(2t) = 50/3. Take ln of both sides: ln(e^(2t)) = ln(50/3). The left side simplifies to 2t, so 2t = ln(50/3) and t = (1/2)·ln(50/3). Plug ln(50/3) into this calculator to finish quickly.

How to avoid common ln mistakes

• Don’t use ln on negatives (real math): ln(-5) isn’t a real number.
• Remember ln(1)=0: if you see ln(1) showing anything else, re-check input.
• Log rules help simplify: ln(ab)=ln(a)+ln(b) and ln(a/b)=ln(a)-ln(b).
• Use change-of-base correctly: log_b(x)=ln(x)/ln(b), not ln(b)/ln(x).
• Rounding: keep extra decimals during intermediate steps, round at the end.

If you’re using ln for calculus, you’ll also see properties like d/dx ln(x) = 1/x (for x>0) and the integral ∫(1/x) dx = ln|x| + C. Those relationships are part of why ln is the “default” log in higher math: it plays beautifully with derivatives and integrals.

Frequently Asked Questions

• What does ln stand for?

  ln stands for natural logarithm. It’s the logarithm with base e. When you write ln(x), you’re asking: “What exponent do I raise e to so that I get x?”

• What’s the difference between ln and log?

  ln is specifically base e. The symbol “log” can mean different bases depending on the context: in many math classes log means base 10, while in computer science log often means base 2. This page shows base conversions so you can compare them side-by-side.

• Why is ln(x) undefined for x ≤ 0?

  Because e^y is always positive for real y. There is no real exponent y that makes e^y equal 0 or a negative number. (Complex logarithms exist, but require complex numbers.)

• How do I compute log base b using ln?

  Use the change-of-base formula: log_b(x) = ln(x) / ln(b). Choose “Custom base b” and this calculator will apply it automatically.

• What does a negative ln value mean?

  If ln(x) is negative, then x is between 0 and 1. For example ln(0.5) ≈ -0.6931 because e^-0.6931 ≈ 0.5.

• Is ln used in finance?

  Yes. ln shows up in continuous compounding and in growth/return calculations. For example, continuously compounded return is often modeled with ln(P₂/P₁). (Always follow the conventions used in your course.)

Think of ln as “undoing multiplication”

Exponentials turn addition into multiplication: when you multiply by a constant factor repeatedly, you’re effectively adding exponents. Logs reverse that process. That’s why log rules exist: ln(ab)=ln(a)+ln(b). Multiplying numbers becomes adding log values.

This also explains why ln is such a powerful “compression” tool. Large ranges of positive numbers become manageable ranges of log values. That’s why you’ll see logs in pH, decibels, and information theory— all of them benefit from turning huge multiplicative differences into smaller additive ones.

If you want one mental anchor: ln is the scale of exponents. It’s like asking, “How many e-multiplications did it take to get here?”

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