---
title: "Logarithm — AP Precalculus Definition & Exam Guide"
description: "A logarithm log_b(c) is the exponent base b needs to produce c. It's the inverse of exponentiation and the key to Topic 2.9 and solving exponential equations."
canonical: "https://fiveable.me/ap-pre-calc/key-terms/logarithm"
type: "key-term"
subject: "AP Pre-Calculus"
unit: "Unit 2"
---

# Logarithm — AP Precalculus Definition & Exam Guide

## Definition

In AP Precalculus, a logarithm answers the question "what exponent do I need?" Formally, log_b(c) = a if and only if b^a = c, where b > 0 and b ≠ 1. It is the inverse of exponentiation, which is why logs are the tool for undoing exponential functions (Topic 2.9).

## What It Is

A logarithm is an exponent in disguise. The expression log_b(c) asks one question. To what power do I raise the [base](/ap-pre-calc/unit-2/inverses-exponential-functions/study-guide/7mdx6zi19alJ4hK3 "fv-autolink") b to get c? The CED states this as a biconditional in 2.9.A.1: log_b(c) = a if and only if b^a = c, with the conditions b > 0 and b ≠ 1. So log_2(8) = 3 because 2^3 = 8. Every log statement is just an exponential statement read backwards.

When no base is written, AP Precalc assumes base 10 (the [common logarithm](/ap-pre-calc/unit-2/logarithmic-expressions/study-guide/OcOVToNeiQNxP11V "fv-autolink")). Some log values you can evaluate by hand because the answer is a nice exponent, like log_5(125) = 3. Others, like log(7), aren't clean powers, so you estimate them with your calculator (2.9.A.2). Logs also power logarithmic scales like the Richter scale, where moving one unit on the scale means multiplying the actual value by the base, not adding to it.

## Why It Matters

Logarithms live in [Unit 2](/ap-pre-calc/unit-2 "fv-autolink"): Exponential and Logarithmic Functions, specifically Topic 2.9, where learning objective 2.9.A asks you to evaluate logarithmic expressions. But the payoff goes way beyond evaluating log_3(81). Logs are how you solve for an unknown exponent, which makes them essential for the exponential modeling problems that dominate Unit 2 FRQs. If a population doubles every 5 years and you need to know when it hits a million, the variable is in the exponent, and a logarithm is the only way to get it down. Logs also underlie logarithmic scales and [semi-log plots](/ap-pre-calc/unit-2/semi-log-plots/study-guide/sGsPkQ8aoU7UKBna "fv-autolink") later in the unit, where exponential data gets straightened into a line.

## Connections

### Exponential Functions (Unit 2)

Logs and exponentials are the same relationship read in opposite directions. b^a = c tells you the [output](/ap-pre-calc/unit-1/change-tandem/study-guide/eQFiTo22fpkDFsnj "fv-autolink") from an exponent; log_b(c) = a recovers the exponent from the output. The 2025 FRQ Q1 setup, a table of values halving each step, is exactly the kind of exponential pattern logs let you solve algebraically.

### Inverse Functions (Unit 2)

The logarithmic function f(x) = log_b(x) is defined as the inverse of g(x) = b^x. That's why their graphs are reflections over y = x and why the [domain](/ap-pre-calc/key-terms/domain "fv-autolink") of one is the range of the other. If you understand inverses from earlier in Unit 2, the log function isn't a new idea, it's a familiar one applied to exponentials.

### Logarithmic Scales and Semi-Log Plots (Unit 2)

On a log scale, each unit step multiplies the value by the base instead of adding a constant. That's why a 7.0 earthquake isn't "1.5 more" than a 5.5, it's 10^1.5 (about 31.6) times more intense. Semi-log plots use this trick to turn exponential curves into straight lines.

### Solving Exponential and Logarithmic Equations (Unit 2)

Equations like log(x) + log(x + 15) = 1 get solved by converting between log form and exponential form, then checking the domain. Logs only accept positive inputs, so a candidate solution that makes any log argument negative gets thrown out. That domain check is a classic point-loser.

## On the AP Exam

Logarithms show up in multiple-choice questions that test the definition directly (evaluate log_2(128) using the fact that logs are exponents), and in applied stems about logarithmic scales, like computing how many times more intense a 7.0 earthquake is than a 5.5 on the base-10 Richter scale. The log-scale questions reward one specific insight from 2.9.A.3: each unit on the scale is a factor of the base, so you compute 10^(7.0 - 5.5), not 7.0 - 5.5. On the free-response side, logarithms appear when an exponential model needs solving for time or an unknown exponent, as in the 2025 FRQ Q1 setup built on an exponential table. Expect to convert between log form and exponential form fluently, evaluate clean logs by hand, and know when a calculator estimate is the move.

## logarithm vs Exponent

A logarithm isn't a different kind of number, it IS an exponent. The confusion comes from notation. In 2^3 = 8, the exponent 3 sits visibly in the expression. In log_2(8) = 3, that same 3 is now the answer. Students often treat log_b(c) as some new operation to memorize, but it's just the question "which exponent?" If you can rewrite every log equation as its exponential twin (log_b c = a means b^a = c), most log problems collapse into exponent problems you already know.

## Key Takeaways

- log_b(c) = a if and only if b^a = c, with b > 0 and b ≠ 1, which means a logarithm is literally just an exponent.
- When no base is written, log means log base 10 (the common logarithm) on the AP Precalc exam.
- Evaluate clean logs by converting to exponential form, like log_5(125) = 3 because 5^3 = 125, and use technology to estimate messy ones.
- On a logarithmic scale, each unit step multiplies the value by the base, so a 7.0 earthquake is 10^1.5 ≈ 31.6 times more intense than a 5.5, not 1.5 times.
- Logarithms are the tool for solving exponential equations because they pull the variable down out of the exponent.
- Log arguments must be positive, so always check that your solutions keep every log input greater than zero.

## FAQs

### What is a logarithm in AP Precalculus?

A logarithm log_b(c) is the exponent the base b must be raised to in order to get c. The CED defines it in 2.9.A.1 as log_b(c) = a if and only if b^a = c, where b > 0 and b ≠ 1.

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

No. Plain "log" with no base means base 10 (the common logarithm), while "ln" means the [natural logarithm](/ap-pre-calc/key-terms/natural-logarithm "fv-autolink") with base e. Both follow the same definition, just with different bases.

### How is a logarithm different from an exponential function?

They're inverses of each other. An [exponential function](/ap-pre-calc/unit-2/exponential-logarithmic-equations-inequalities/study-guide/Mor8iJ1w4VWPX8Wi "fv-autolink") takes an exponent and gives you an output (2^3 = 8), while a logarithm takes the output and gives you back the exponent (log_2(8) = 3). Graphically they're reflections over the line y = x.

### Why is a 7.0 earthquake so much stronger than a 5.5?

The Richter scale is logarithmic with base 10, so each unit on the scale multiplies intensity by 10. The difference of 1.5 units means the 7.0 quake is 10^1.5, or about 31.6 times, more intense, not 1.5 times more.

### Can a logarithm have a negative input or a base of 1?

No to both. The argument of a log must be positive, and the base must be positive and not equal to 1, since 1 raised to any power is just 1. That's why solving log equations always ends with a domain check that can eliminate solutions.

## Related Study Guides

- [2.9 Logarithmic Expressions](/ap-pre-calc/unit-2/logarithmic-expressions/study-guide/OcOVToNeiQNxP11V)

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