---
title: "Natural Logarithm — AP Precalculus Definition & Guide"
description: "The natural logarithm ln x is the log with base e, the inverse of e^x. In AP Precalc Unit 2, it solves exponential models and powers change of base."
canonical: "https://fiveable.me/ap-pre-calc/key-terms/natural-logarithm"
type: "key-term"
subject: "AP Pre-Calculus"
unit: "Unit 2"
---

# Natural Logarithm — AP Precalculus Definition & Guide

## Definition

The natural logarithm, written ln x, is the logarithmic function with base e (so ln x = log_e x). In AP Precalculus Unit 2, it's the inverse of the exponential function e^x, which makes it the go-to tool for solving exponential growth and decay models for time.

## What It Is

The natural logarithm is just a [logarithm](/ap-pre-calc/key-terms/logarithm "fv-autolink") with a special [base](/ap-pre-calc/unit-2/inverses-exponential-functions/study-guide/7mdx6zi19alJ4hK3 "fv-autolink"). Instead of base 10 or base 2, it uses base e (about 2.718), and we abbreviate log_e x as ln x. Everything you know about logs still applies. The product property, power property, and change of base formula all work exactly the same way with ln.

Why does e get its own button on your calculator? Because so many real situations involving continuous [proportional growth](/ap-pre-calc/unit-2/logarithmic-function-context-data-modeling/study-guide/viEiJjFy5ba6uEyw "fv-autolink") (radioactive decay, continuously compounded interest, population models) are written with e as the base, like A(t) = A₀e^(-kt). Since ln x and e^x are inverse functions, ln undoes e. When you need to pull a variable out of an exponent in one of these models, taking the natural log of both sides is the move. That's the whole reason ln shows up constantly in Topics 2.12 and 2.14.

## Why It Matters

Natural log lives in [Unit 2](/ap-pre-calc/unit-2 "fv-autolink") (Exponential and Logarithmic Functions), specifically Topics 2.12 and 2.14. Learning objective 2.12.A asks you to rewrite [logarithmic expressions](/ap-pre-calc/unit-2/logarithmic-expressions/study-guide/OcOVToNeiQNxP11V "fv-autolink") in equivalent forms, and the change of base property means any logarithm can be rewritten using ln, which is exactly what your calculator often expects. Learning objective 2.14.A asks you to construct logarithmic function models, and the essential knowledge there is that logarithmic functions are inverses of exponential functions. Since e is the most common base for exponential models, ln is the inverse you'll reach for most. If you can't comfortably use ln to solve A₀e^(kt) = some value for t, the modeling questions in Unit 2 become guesswork.

## Connections

### Exponential function models with base e (Unit 2)

Models like A(t) = A₀e^(-kt) for radioactive decay are solved by taking ln of both sides. Because ln and e^x are inverses, ln(e^(-kt)) collapses to -kt, and suddenly t is solvable. Natural log is the key that unlocks the exponent.

### Change of base and log properties (Unit 2, Topic 2.12)

The [change of base property](/ap-pre-calc/unit-2/logarithmic-function-manipulation/study-guide/tBicQgqFwNoesetP "fv-autolink") lets you rewrite any log in terms of ln, so log_b x = ln x / ln b. This is how you evaluate a log base 1.06 or base 0.97 on a calculator that only has ln and log buttons.

### [Horizontal dilation (Unit 2)](/ap-pre-calc/key-terms/horizontal-dilation)

The product property says ln(kx) = ln k + ln x. In graph language, a [horizontal dilation](/ap-pre-calc/key-terms/horizontal-dilation "fv-autolink") of the natural log graph is the same thing as a vertical translation. That equivalence is pure 2.12.A.1 and it shows up in MCQs about transformed log graphs.

### Real zero of a log model (Unit 2, Topic 2.14)

Every natural log function f(x) = ln x has its [real zero](/ap-pre-calc/key-terms/real-zero "fv-autolink") at x = 1, since ln 1 = 0. When you build a log model from data, that zero (possibly shifted by transformations) anchors where the model crosses the x-axis.

## On the AP Exam

You won't get a question that just asks "define ln." Instead, natural log is a tool you're expected to deploy. Typical tasks include solving an exponential model like A(t) = A₀e^(-kt) for the decay constant or for time (for example, finding when a radioactive sample drops below a threshold, given that 30% remains after 50 years), rewriting expressions like pH = -log₁₀[H⁺] using natural logs via change of base, and finding doubling or growth times in compound interest models like A(t) = P(1+r)^t by taking ln of both sides. MCQs also test the log properties applied to ln, such as recognizing that ln(kx) = ln k + ln x means a horizontal dilation equals a vertical shift. Show the algebra cleanly. The free-response rubric rewards the step where you take ln of both sides and simplify using inverse properties.

## natural logarithm vs common logarithm (log x, base 10)

Both are logarithms, just with different bases. The common log uses base 10 and pairs naturally with powers of 10 (like the pH scale). The natural log uses base e and pairs with continuous growth models like e^(kt). They're connected by change of base, so log₁₀ x = ln x / ln 10. On the exam, match the log to the base of the exponential in the problem. If the model uses e, use ln; if it uses 10, log₁₀ is cleaner. Either one technically works, since change of base converts between them.

## Key Takeaways

- The natural logarithm ln x means log base e of x, where e ≈ 2.718, and it is the inverse function of e^x.
- To solve an exponential model like A(t) = A₀e^(-kt) for time, take the natural log of both sides so the exponent comes down and you can isolate t.
- All the standard log properties apply to ln, including the product property ln(xy) = ln x + ln y and the power property ln(x^n) = n ln x.
- The change of base property, log_b x = ln x / ln b, lets you evaluate a logarithm of any base using only the ln button on your calculator.
- Graphically, ln(kx) = ln k + ln x means a horizontal dilation of the natural log graph is the same as a vertical translation, which is tested directly under 2.12.A.
- Because ln 1 = 0, the basic natural log function has its real zero at x = 1, a fact you can use to anchor or check a logarithmic model.

## FAQs

### What is the natural logarithm in AP Precalculus?

It's the logarithm with base e (about 2.718), written ln x, so ln x = log_e x. In Unit 2 it works as the inverse of e^x, which is why you use it to solve exponential growth and decay models.

### Is ln the same as log on the AP exam?

No. On calculators and the exam, log usually means log base 10 (the common log), while ln always means log base e. They give different numbers for the same input, but change of base converts between them: log₁₀ x = ln x / ln 10.

### Why do we use ln instead of log base 10 for decay problems?

Because decay models are typically written with base e, like A(t) = A₀e^(-kt). Taking ln of both sides cancels the e cleanly, since ln(e^(-kt)) = -kt. Using log₁₀ would still work but leaves messier algebra.

### Do the log properties work for natural logs too?

Yes, all of them. The product property ln(xy) = ln x + ln y, the power property ln(x^n) = n ln x, and change of base all hold because ln is just a logarithm with base e. Topic 2.12 tests these properties regardless of base.

### What is ln(e) and ln(1)?

ln(e) = 1 because e¹ = e, and ln(1) = 0 because e⁰ = 1. These two values come straight from the inverse relationship between ln x and e^x, and they're worth memorizing for quick MCQ checks.

## Related Study Guides

- [2.12 Logarithmic Function Manipulation](/ap-pre-calc/unit-2/logarithmic-function-manipulation/study-guide/tBicQgqFwNoesetP)

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