Welcome to the World of Logarithms!

If you have ever looked at an equation like \(2^x = 8\) and known the answer was \(3\), you already understand the heart of a logarithm. Logarithms are essentially "exponent detectives"—their entire job is to tell you what power a number was raised to. While they might look intimidating at first, they are just a different way of writing the exponential relationships you already know.

In this chapter, we focus on Topic 2.9: Logarithmic Expressions. We will learn how to translate between exponential and logarithmic forms and how to evaluate these expressions without needing a calculator.


1. What is a Logarithm?

At its simplest, a logarithm is the inverse of an exponent. If an exponential expression tells us the result of raising a base to a power, a logarithmic expression tells us the power itself.

The Definition:
The expression \(\log_b(c) = a\) is equivalent to the exponential equation \(b^a = c\).

Think of it as a loop:

  • \(b\) is the base (it stays the base in both forms).
  • \(a\) is the exponent (the answer to the log).
  • \(c\) is the argument (the result of the power).

Analogy: The Exponent Detective
Imagine you see the expression \(\log_2(16)\). The logarithm is asking a question: "Hey, Detective! What exponent do I put on a base of 2 to get 16?" Since \(2^4 = 16\), the detective reports back: "The answer is 4!" Therefore, \(\log_2(16) = 4\).


2. Rewriting Expressions: The Core Skill

To succeed in AP Precalculus, you must be able to switch between forms fluently. This is a key part of Skill 1.B: expressing expressions in analytically equivalent forms.

From Exponential to Logarithmic

If you have \(5^3 = 125\), the base is \(5\) and the exponent is \(3\).
In log form: \(\log_5(125) = 3\).

From Logarithmic to Exponential

If you have \(\log_7(49) = 2\), the base is \(7\) and the exponent is \(2\).
In exponential form: \(7^2 = 49\).

Quick Review Box:
\(b^a = c \iff \log_b(c) = a\)
Mnemonic: "The base stays the base, and the other two switch places!"


3. Evaluating Logarithms Step-by-Step

On Section I Part A of the AP Exam, you won't have a calculator. You need to evaluate expressions like \(\log_3\left(\frac{1}{9}\right)\) by hand. Here is how:

Step 1: Set the expression equal to \(x\).
\(\log_3\left(\frac{1}{9}\right) = x\)

Step 2: Rewrite it in exponential form.
\(3^x = \frac{1}{9}\)

Step 3: Express both sides with the same base if possible.
Since \(9 = 3^2\), then \(\frac{1}{9} = 3^{-2}\).
So, \(3^x = 3^{-2}\).

Step 4: Solve for \(x\).
\(x = -2\)

Common Mistakes to Avoid:
Don't confuse \(\log_9(3)\) with \(\log_3(9)\)!
\(\log_3(9) = 2\) (because \(3^2 = 9\))
\(\log_9(3) = \frac{1}{2}\) (because \(9^{1/2} = \sqrt{9} = 3\))


4. Special Logarithms: Common and Natural

In mathematics, some bases are so common that we have "shortcuts" for writing them. You will see these frequently in Topic 2.11 (Logarithmic Functions) and Topic 2.14 (Modeling Contexts).

The Common Logarithm (Base 10)

When you see a logarithm written without a base, it is assumed to be base 10.
\(\log(x)\) is the same as \(\log_{10}(x)\).
Example: \(\log(100) = 2\) because \(10^2 = 100\).

The Natural Logarithm (Base \(e\))

The number \(e\) (approx. \(2.718\)) is a fundamental constant in calculus. A logarithm with base \(e\) is called the Natural Logarithm.
\(\ln(x)\) is the same as \(\log_e(x)\).
Example: \(\ln(e^5) = 5\) because the exponent needed on base \(e\) to get \(e^5\) is simply \(5\).

Did you know? The notation "ln" comes from the Latin logarithmus naturalis.


5. Important Restrictions

Logarithms have specific "rules of the road" regarding what numbers you can use. These are vital for determining the domain of functions later on.

  • The Argument must be positive: In \(\log_b(c)\), \(c\) must be greater than zero (\(c > 0\)). You cannot take the log of a negative number or zero in the real number system because there is no power you can raise a positive base to that results in a negative value.
  • The Base must be positive and not 1: We require \(b > 0\) and \(b \neq 1\). If the base were \(1\), raising it to any power would always just stay \(1\), which isn't very useful for modeling change!

Key Takeaways for Topic 2.9

1. Definition: \(\log_b(c) = a\) means \(b^a = c\).
2. Purpose: A logarithm finds the missing exponent.
3. Evaluation: To find the value, ask: "What power do I raise the base to in order to get the argument?"
4. Notation: No base written means base \(10\); \(\ln\) means base \(e\).
5. Next Steps: Understanding these expressions is the foundation for Topic 2.10: Inverses of Exponential Functions and Topic 2.12: Logarithmic Function Manipulation.