SOLID STATE PRESS
← Back to catalog
Logarithmic Functions cover
Coming soon
Coming soon to Amazon
This title is in our publishing queue.
Browse available titles
Mathematics

Logarithmic Functions

A High School and Early College Primer

Logarithms stop a lot of students cold. The notation looks strange, the rules feel arbitrary, and most textbooks bury the core idea under pages of definitions before anything clicks. If you have a precalculus test coming up, an Algebra II unit to finish, or a kid who keeps asking "what even is a log?", this guide gets straight to the point.

**TLDR: Logarithmic Functions** covers everything a high school or early college student needs: what a logarithm actually means (and why it's just an exponent in disguise), how to evaluate logs by inspection and with the change-of-base formula, and how to apply the product, quotient, and power rules for expanding and condensing expressions. From there it walks through graphing log functions — domain restrictions, vertical asymptotes, and transformations — then tackles solving exponential and logarithmic equations, including how to spot and discard extraneous solutions. The final section connects all of it to real contexts: pH, decibels, the Richter scale, compound interest, and half-life.

This is a focused algebra 2 logarithmic functions study guide, not a textbook. It's 10–20 pages of clear explanation, worked examples, and exactly the misconception-busting notes your teacher wishes they had time to say out loud. No filler, no fluff — just the concepts, the patterns, and the confidence to use them.

If you need to get up to speed on logs fast, start here.

What you'll learn
  • Translate fluently between exponential and logarithmic form
  • Evaluate common, natural, and arbitrary-base logarithms by hand and with the change-of-base formula
  • Apply the product, quotient, and power properties to expand and condense log expressions
  • Graph logarithmic functions and identify domain, range, asymptotes, and transformations
  • Solve exponential and logarithmic equations and apply them to growth, decay, pH, and decibel problems
What's inside
  1. 1. What a Logarithm Actually Is
    Introduces the logarithm as the inverse of exponentiation and builds intuition through small numerical examples.
  2. 2. Evaluating Logs and the Change-of-Base Formula
    Shows how to evaluate logs by inspection, by rewriting as exponentials, and using change-of-base for non-standard bases.
  3. 3. Properties of Logarithms
    Covers the product, quotient, and power rules, plus expanding and condensing log expressions.
  4. 4. Graphs, Domain, and Transformations
    Examines the shape of logarithmic graphs, their domain restrictions, vertical asymptotes, and standard transformations.
  5. 5. Solving Exponential and Logarithmic Equations
    Walks through the main techniques for solving equations involving exponentials and logs, including extraneous solutions.
  6. 6. Where Logarithms Show Up
    Connects logs to real applications: pH, decibels, the Richter scale, compound interest, and half-life problems.
Published by Solid State Press
Logarithmic Functions cover
TLDR STUDY GUIDES

Logarithmic Functions

A High School and Early College Primer
Solid State Press

Logarithmic Functions

A High School and Early College Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Who This Book Is For

If you are a high school student who needs logarithms explained clearly — maybe you are working through Algebra 2 logarithmic functions for the first time, or you are deep into a precalculus course and the rules are starting to blur — this book is for you. It is also for the parent trying to help a kid with logs and exponentials the night before a test, or the tutor who needs a clean, fast review.

This is a short math study guide built specifically for the precalculus and Algebra II level. It covers what logarithms are and how to evaluate them, the log properties — product, quotient, and power rules — and how to solve logarithmic equations step by step. Graphs, transformations, the change-of-base formula, and real-world applications are all included. About 15 pages, no filler.

Read it straight through, work every example as you go, then use the problem set at the end to confirm you have it.

Contents

  1. 1 What a Logarithm Actually Is
  2. 2 Evaluating Logs and the Change-of-Base Formula
  3. 3 Properties of Logarithms
  4. 4 Graphs, Domain, and Transformations
  5. 5 Solving Exponential and Logarithmic Equations
  6. 6 Where Logarithms Show Up
Chapter 1

What a Logarithm Actually Is

A logarithm answers one question: what exponent do I need? More precisely, the logarithm base $b$ of a number $x$ is the exponent you must raise $b$ to in order to get $x$.

That single sentence is the whole idea. Everything else in this topic is a consequence of it.

Write it symbolically:

$\log_b x = e \quad \text{means exactly the same thing as} \quad b^e = x$

The expression $\log_b x$ is read "log base $b$ of $x$." The number $b$ is the base, and $x$ is the argument. The equation $b^e = x$ is called exponential form; the equation $\log_b x = e$ is called logarithmic form. These two forms carry identical information — they are just different ways of writing the same relationship.

Think of it this way: exponentiation takes a base and an exponent and produces a result. Logarithms run that process backward — they take the base and the result and produce the exponent. If exponentiation is a machine that goes $(\text{base}, \text{exponent}) \to \text{result}$, then a logarithm is the same machine run in reverse.

Building Intuition with Small Numbers

Start with powers of $2$ that you already know:

$2^1 = 2, \quad 2^2 = 4, \quad 2^3 = 8, \quad 2^4 = 16, \quad 2^{10} = 1024$

Translating each into logarithmic form:

$\log_2 2 = 1, \quad \log_2 4 = 2, \quad \log_2 8 = 3, \quad \log_2 16 = 4, \quad \log_2 1024 = 10$

Every time you see $\log_2(\text{something})$, you are simply asking: "What power of $2$ gives me that something?"

Example. Evaluate $\log_3 81$.

Solution. Ask: "What power of $3$ equals $81$?" Since $3^4 = 81$, the answer is $4$. So $\log_3 81 = 4$.

Example. Evaluate $\log_5 \tfrac{1}{25}$.

Solution. Ask: "What power of $5$ equals $\tfrac{1}{25}$?" Since $5^{-2} = \tfrac{1}{25}$, the answer is $-2$. So $\log_5 \tfrac{1}{25} = -2$.

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters in roughly fifteen pages — readable in one sitting.

Coming soon to Amazon