---
title: "Properties of Logarithms — AP Precalc Definition & Guide"
description: "Properties of logarithms (product, quotient, power, change of base) are the rules that let you rewrite and solve log equations in AP Precalc Topic 2.13."
canonical: "https://fiveable.me/ap-pre-calc/key-terms/properties-of-logarithms"
type: "key-term"
subject: "AP Pre-Calculus"
unit: "Unit 2"
---

# Properties of Logarithms — AP Precalc Definition & Guide

## Definition

Properties of logarithms are the rules (product, quotient, power, and change of base) that let you rewrite logarithmic expressions, used in AP Precalculus Topic 2.13 to solve exponential and logarithmic equations and inequalities, then check for extraneous solutions.

## What It Is

Properties of logarithms are the algebraic rules that tell you how logs interact with multiplication, division, and exponents. The product rule says log_b(MN) = log_b(M) + log_b(N). The [quotient](/ap-pre-calc/unit-1/equivalent-representations-polynomial-rational-expressions/study-guide/NRzwc7vjmULoqIyP "fv-autolink") rule says log_b(M/N) = log_b(M) - log_b(N). The power rule says log_b(M^p) = p·log_b(M). The [change of base formula](/ap-pre-calc/key-terms/change-of-base-formula "fv-autolink"), log_b(M) = log_c(M)/log_c(b), lets you rewrite a log in any base you want, usually base 10 or base e so your calculator can handle it.

Here's the intuition that makes all of them click. Logarithms are exponents, so every log property is just an exponent property wearing a disguise. Multiplying powers means adding exponents, so multiplying inside a log means adding logs. In the CED, these properties show up in [Topic 2.13](/ap-pre-calc/unit-2/exponential-logarithmic-equations-inequalities/study-guide/Mor8iJ1w4VWPX8Wi "fv-autolink"), where they work alongside properties of exponents and the inverse relationship between exponential and logarithmic functions to solve equations analytically. The CED also highlights one specific rewrite worth memorizing, b^x = c^((log_c b)(x)), which converts any exponential base into any other base.

## Why It Matters

This lives in [Unit 2](/ap-pre-calc/unit-2 "fv-autolink") (Exponential and Logarithmic Functions), specifically Topic 2.13, and directly supports learning objective 2.13.A, solving exponential and logarithmic equations and inequalities. The essential knowledge is explicit that properties of logarithms, [properties of exponents](/ap-pre-calc/key-terms/properties-of-exponents "fv-autolink"), and the exponential-log inverse relationship are your toolkit for solving these equations. They also support 2.13.B, since constructing the inverse of f(x) = a log_b(x-h) + k means undoing log operations step by step. On the exam, you often can't solve a log equation until you've condensed several logs into one or expanded one log into pieces, so these properties are usually step one of the algebra, not an optional shortcut. And because Unit 2 makes up a big chunk of the AP Precalculus exam, fluency here pays off across many questions.

## Connections

### [Properties of Exponents (Unit 2)](/ap-pre-calc/key-terms/properties-of-exponents)

Every log property is an exponent property in reverse. Adding exponents when you multiply powers becomes adding logs when you multiply arguments. If you forget a log rule on the exam, you can rebuild it from the matching exponent rule.

### [Change of Base Formula (Unit 2)](/ap-pre-calc/key-terms/change-of-base-formula)

Change of [base](/ap-pre-calc/unit-2/inverses-exponential-functions/study-guide/7mdx6zi19alJ4hK3 "fv-autolink") is technically one of the properties, but it earns its own spotlight because it's how you evaluate something like log_3(7) with a calculator that only knows base 10 and base e. It's also behind the CED rewrite b^x = c^((log_c b)(x)).

### [Extraneous Solutions (Unit 2)](/ap-pre-calc/key-terms/extraneous-solutions)

Condensing logs can quietly change the [domain](/ap-pre-calc/key-terms/domain "fv-autolink") of an equation. A solution that works in the condensed version might make an original log argument negative or zero, and the CED specifically tells you to check for these extraneous solutions after solving.

### [Exponential Equation (Unit 2)](/ap-pre-calc/key-terms/exponential-equation)

When the variable is stuck in an exponent, like 5^x = 12, taking a log of both sides and using the power rule pulls the variable down where you can solve for it. Log properties are the bridge between exponential equations and linear algebra you already know.

## On the AP Exam

Properties of logarithms show up in multiple-choice questions that ask you to condense or expand a logarithmic expression, solve a log or exponential equation analytically, or recognize the exact form of a solution. A classic setup gives you an equation like e^(2x) - 5e^x + 6 = 0, which factors like a quadratic in e^x, and the answer choices are exact values like ln 2 and ln 3 rather than decimals. A numeric solver gives you decimals, but the exam wants the analytical form, and log properties get you there. After solving, always check whether a solution makes any original log argument zero or negative, because the CED expects you to rule out extraneous solutions. On the calculator-allowed sections, change of base is your tool for evaluating logs in unusual bases.

## properties of logarithms vs properties of exponents

They're mirror images, not the same thing. Properties of exponents tell you how to manipulate powers (b^m · b^n = b^(m+n)), while properties of logarithms tell you how to manipulate logs (log adds when arguments multiply). The most common exam error is mixing the directions, like writing log(M + N) = log M + log N. That's false. Logs turn multiplication into addition, never addition into addition.

## Key Takeaways

- The product rule turns multiplication inside a log into addition of logs, the quotient rule turns division into subtraction, and the power rule lets you pull an exponent out front as a coefficient.
- Logarithms are exponents, so every log property is just an exponent property flipped around, which is why the two rule sets always travel together in Topic 2.13.
- The change of base formula, log_b(M) = log_c(M)/log_c(b), lets you evaluate any log with a calculator and supports the CED rewrite b^x = c^((log_c b)(x)).
- After using log properties to solve an equation, plug your answers back into the original equation and reject any extraneous solution that makes a log argument zero or negative.
- There is no property for log(M + N), so never split a log of a sum into a sum of logs.

## FAQs

### What are the properties of logarithms in AP Precalculus?

They are the product rule (log of a product equals sum of logs), quotient rule (log of a quotient equals difference of logs), power rule (exponents pull out as coefficients), and the change of base formula. They appear in Topic 2.13 as tools for solving exponential and logarithmic equations under learning objective 2.13.A.

### Is log(a + b) the same as log(a) + log(b)?

No, and this is one of the most common errors on the exam. The product rule says log(ab) = log(a) + log(b), so logs turn multiplication into addition. There is no rule that simplifies the log of a sum.

### How are properties of logarithms different from properties of exponents?

They're inverses of each other. [Exponent rules](/ap-pre-calc/key-terms/exponent-rules "fv-autolink") say b^m · b^n = b^(m+n), and the matching log rule says log_b(MN) = log_b(M) + log_b(N). Same relationship, opposite direction, and the CED expects you to use both together when solving equations.

### Do I need the change of base formula on the AP Precalculus exam?

Yes. Calculators typically only compute base-10 and base-e logs, so evaluating something like log_3(7) requires log(7)/log(3). It also powers the CED's exponential rewrite b^x = c^((log_c b)(x)).

### Why do log equations have extraneous solutions?

Log arguments must be positive, but condensing logs with the product or quotient rule can hide that restriction. A value that solves the condensed equation might make an original argument negative or zero, so the CED requires you to check every solution against the original equation.

## Related Study Guides

- [2.13 Exponential and Logarithmic Equations and Inequalities](/ap-pre-calc/unit-2/exponential-logarithmic-equations-inequalities/study-guide/Mor8iJ1w4VWPX8Wi)

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