The Natural Logarithm and its Rules

Short answer: The natural logarithm, ln(x), is the logarithm with base e, where e ≈ 2.718. It answers one question: e to what power gives x? It follows the same rules as every logarithm: ln(ab) = ln a + ln b, ln(a ÷ b) = ln a − ln b, and ln(aⁿ) = n · ln a. Also, ln 1 = 0, ln e = 1, and ln(eˣ) = x.

Those few rules do almost all the work in precalculus, calculus and physics. Learn them once, with the reason each one is true, and exponential equations stop being guesswork.

Key points

• Base e: ln(x) means log base e, so ln(x) = y exactly when eʸ = x.
• Undoes eˣ: ln and eˣ are inverse functions: ln(eˣ) = x and e^(ln x) = x.
• Same rules as log: Product, quotient, power and change of base work the same in every base.
• Positive inputs only: ln(x) is defined only for x > 0; ln 0 and ln of a negative are undefined.
• Two values to memorize: ln 1 = 0 and ln e = 1.
• Rules come from exponents: Each log rule is an exponent rule read backwards.

What is the natural logarithm?

The natural logarithm is the logarithm whose base is the number e = 2.71828… (an irrational number, like π). By definition, ln(x) = logₑ(x).

A logarithm is an exponent. Asking for ln(20) is asking “e raised to what power equals 20?” The answer is about 2.996, because e^2.996 ≈ 20.

You may already know the common logarithm, log(x), which is log base 10 and undoes y = 10ˣ. The natural logarithm plays the same role for y = eˣ:

• ln(eˣ) = x for every real number x.
• e^(ln x) = x for every x > 0.

Because eˣ is always positive, ln(x) only accepts positive inputs. Its graph crosses the x-axis at x = 1, is negative between 0 and 1, and grows slowly but without limit as x increases.

If logarithms still feel like an arbitrary rule, start with Part 1 of our Understanding Logarithms lessons, on what a logarithm actually asks.

What are the rules of the natural log?

Here is every ln rule you need, with a numerical check for each. The same rules hold for log base 10 or any other base (OpenStax Precalculus, Logarithmic Properties).

The change of base rule also works the other way: ln a = log_b(a) ÷ log_b(e) for any base b. That is how a calculator with only a “log” key can still find a natural log.

What are the most common ln mistakes?

Most lost points on log problems come from “rules” that look right but are false. Watch for these:

• ln(a + b) is not ln a + ln b. There is no rule for the log of a sum. ln(2 + 3) = ln 5 ≈ 1.609, but ln 2 + ln 3 ≈ 1.792.
• ln a ÷ ln b is not ln(a ÷ b). The quotient rule turns the log of a quotient into a difference, not a quotient of logs.
• (ln a)² is not 2 ln a. The power rule only applies when the exponent is inside the log: ln(a²) = 2 ln a.
• ln of zero or a negative number is undefined. Any solution that would put a negative inside ln must be thrown out.

Tutor tip: Before using a log rule, ask whether the expression inside the ln is a product, a quotient or a power. If it is a sum or difference, no rule applies, so simplify inside first.

How do you use the natural log rules? Worked examples

Example 1: Expand ln(x³y ÷ z)

1. Quotient rule: ln(x³y ÷ z) = ln(x³y) − ln z.
2. Product rule: ln(x³y) = ln(x³) + ln y.
3. Power rule: ln(x³) = 3 ln x.
4. Result: ln(x³y ÷ z) = 3 ln x + ln y − ln z.

Example 2: Condense 2 ln x + ln 5 − ln 3

1. Power rule in reverse: 2 ln x = ln(x²).
2. Product rule in reverse: ln(x²) + ln 5 = ln(5x²).
3. Quotient rule in reverse: ln(5x²) − ln 3 = ln(5x² ÷ 3).

Example 3: Solve 3e^(2x) − 1 = 14

1. Isolate the exponential: 3e^(2x) = 15, so e^(2x) = 5.
2. Take ln of both sides: ln(e^(2x)) = ln 5, so 2x = ln 5.
3. Divide by 2: x = ln 5 ÷ 2 ≈ 1.609 ÷ 2 ≈ 0.805.
4. Check: e^(2 × 0.805) = e^1.609 ≈ 5, and 3 × 5 − 1 = 14.

Example 4: Solve ln x + ln(x − 2) = ln 8

1. Product rule: ln(x(x − 2)) = ln 8.
2. Because ln is one-to-one, x(x − 2) = 8, so x² − 2x − 8 = 0.
3. Factor: (x − 4)(x + 2) = 0, so x = 4 or x = −2.
4. Check the domain: x = −2 would put a negative inside ln, so reject it.
5. Check x = 4: ln 4 + ln 2 = ln 8. The answer is x = 4.

Example 5: How long does money take to double at 5% continuous interest?

1. Continuous growth follows A = Pe^(rt). Doubling means A = 2P, so 2 = e^(0.05t).
2. Take ln of both sides: ln 2 = 0.05t.
3. Solve: t = ln 2 ÷ 0.05 ≈ 0.6931 ÷ 0.05 ≈ 13.86 years.

Tutor tip: Always substitute your answer back into the original equation. With logs, the check catches both arithmetic slips and “solutions” that fall outside the domain, as in Example 4.

Here is the same idea doing real work. Part 2 of the series solves exponential equations with logarithms from start to finish.

If quadratics like the one in Example 4 are rusty, our guide to the quadratic formula covers every case. Many log errors are really algebra errors, so the five most common algebra mistakes is worth a look too.

Why is e the “natural” base?

Most classes skip this, which makes ln feel arbitrary. The reason is the graph of y = eˣ.

A curve has a slope at each point: the slope of its tangent line there. For y = eˣ, the slope at every point equals the y-value at that point. At x = 1 the height is e and the slope is also e; at x = 0 both are 1.

Apart from constant multiples of it, eˣ is the only function with this property. That is why it describes anything whose rate of change is proportional to its size: continuously compounded interest, population growth, radioactive decay and cooling.

The natural log inherits the same simplicity. In calculus, the derivative of ln x is 1 ÷ x, and ln x equals the area under the curve y = 1 ÷ t from t = 1 to t = x. In fact, e can be defined as the one positive number with ln e = 1 (NIST Digital Library of Mathematical Functions, §4.2).

Where do the natural log rules come from?

“At Alexander Tutoring, we derive every formula. Instead of memorizing a formula and plugging numbers in, students see where it comes from.”

Sean Alexander, founder of Alexander Tutoring

Every log rule is an exponent rule in disguise, because ln and eˣ are inverses. Here is the product rule.

1. Let m = ln a and n = ln b. In exponential form, a = eᵐ and b = eⁿ.
2. Multiply: ab = eᵐ · eⁿ = e^(m + n), using the rule for multiplying powers.
3. Take ln of both sides: ln(ab) = ln(e^(m + n)) = m + n.
4. Substitute back: ln(ab) = ln a + ln b.

The other rules follow the same pattern:

• Quotient rule: a ÷ b = eᵐ ÷ eⁿ = e^(m − n), so ln(a ÷ b) = m − n = ln a − ln b.
• Power rule: aᵏ = (eᵐ)ᵏ = e^(mk), so ln(aᵏ) = mk = k ln a.
• Change of base: start from a = eᵐ and take log base c of both sides: log_c(a) = m · log_c(e), so ln a = log_c(a) ÷ log_c(e). Setting a = x and using ln instead of log_c gives log_b(x) = ln x ÷ ln b.

Tutor tip: If you forget a log rule on a test, rebuild it in 30 seconds from the matching exponent rule. Students who can derive a rule rarely misapply it. For more ways to make formulas stick, see how to memorize math formulas.

Is the natural log on the SAT, ACT or AP Calculus?

• SAT: College Board’s list of SAT Advanced Math skills includes exponential equations but does not list logarithms (College Board, SAT Advanced Math). See which formulas to memorize for the SAT.
• ACT: ACT’s math test description lists logarithmic functions among the function types that may appear (ACT math test description). Our breakdown of math topics on the ACT has more.
• AP Calculus and precalculus: ln is everywhere: derivatives, integrals of 1 ÷ x, and exponential growth and decay. Our guide to AP math classes explains how to prepare.

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