2.9 Logarithmic Expressions

TLDR

A logarithm answers the question "what exponent do I need?" The expression $\log_b c = a$ means exactly the same thing as $b^a = c$, so $\log_2 8 = 3$ because $2^3 = 8$. In AP Precalculus, you evaluate logs by rewriting them in exponential form, and you read logarithmic scales as multiplicative jumps instead of equal steps.

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How to Read Logarithms

To read a logarithm, translate it into an exponent question. The expression $\log_b c$ means "the exponent on base $b$ that gives $c$." So $\log_b c = a$ is equivalent to $b^a = c$.

For Topic 2.9, this rewrite is the main skill. Use it to evaluate exact logarithmic expressions, check that the base is positive and not 1, and remember that the argument must be positive. On logarithmic scales, equal steps represent multiplication by the base, not adding the same amount each time.

Why This Matters for the AP Precalculus Exam

Logarithmic expressions are the bridge between exponents and logarithmic functions, so this skill shows up across Unit 2, which carries a large share of the exam. You will see logs in multiple-choice questions that ask for exact values, and the same idea supports free-response work where you solve equations or interpret models. On the no-calculator parts, you are expected to evaluate clean logs like $\log_3 81$ by hand, so being quick with the exponential rewrite saves time.

Getting comfortable here sets up everything later in the unit: inverses of exponential functions, logarithmic functions and their graphs, log properties, and solving exponential and logarithmic equations.

Key Takeaways

• $\log_b c = a$ is true exactly when $b^a = c$. Switch between these two forms to evaluate any log.
• The base $b$ must satisfy $b > 0$ and $b \neq 1$, and the argument $c$ must be positive.
• $\log c$ with no written base means the common logarithm, base 10. $\ln c$ means the natural logarithm, base $e \approx 2.718$.
• A logarithm is the inverse of an exponential function, so taking a log undoes raising to a power.
• Some logs are exact integers you can find by hand; others need a calculator.
• On a logarithmic scale, each step is a multiplicative change by the base, not a fixed amount.

What a Logarithm Actually Means

The logarithmic expression $\log_b c$ is the "logarithm of $c$ to the base $b$." It represents the exponent the base must be raised to in order to produce $c$.

Formally:

$\log_b c = a \quad \text{if and only if} \quad b^a = c$

where $b > 0$ and $b \neq 1$. In the equation $y = \log_a x$, which is the same as $a^y = x$, the value $y$ is the exponent, $a$ is the base, and $x$ is the argument.

The base cannot be 1, because $1^a$ is always 1, so the log would not have a unique value. The base must also be positive, and the argument $c$ must be positive since a positive base raised to any real power stays positive.

Two named logs come up constantly:

• The common logarithm has base 10 and is usually written $\log c$ with no base shown.
• The natural logarithm has base $e \approx 2.71828$ and is written $\ln c$.

Relating Logs and Exponents

A logarithm is the inverse of an exponential function (the inverse idea from inverse functions). If you have $b^x = c$, you can solve for $x$ by taking the log base $b$ of both sides, which gives $x = \log_b c$.

Some logs come out clean and can be done by hand. Others need technology.

• $\log_2 64$ is easy: rewrite 64 as a power of 2. Since $64 = 2^6$, you get $\log_2 64 = \log_2 2^6 = 6$.
• $\log_2 1000000$ is not a clean power of 2, so you would use a calculator to estimate it.

When you are stuck, ask yourself: "the base raised to what power gives the argument?" That question turns every log evaluation into an exponent you already know.

Logarithmic Scales

A logarithmic scale represents values across a wide range of magnitudes in a compact way. It works differently from the linear scale you have used since earlier math courses.

• On a standard linear scale, the units are equally spaced, and each step represents a fixed increment of the value.
• On a logarithmic scale, each unit represents a multiplicative change by the base of the logarithm.

For example, on a linear scale the marks might be $0, 1, 2, 3, 4, 5, \dots$, while on a base-10 logarithmic scale the marks might be $10^0, 10^1, 10^2, 10^3, 10^4, 10^5, \dots$. Each step is a power of 10, so each move up multiplies by 10.

This is why log scales are handy for data that spans many orders of magnitude. As applications, fields like astronomy, engineering, medicine, and finance use them to display values that would be hard to compare on a linear scale.

How to Use This on the AP Precalculus Exam

MCQ

• For exact-value questions, rewrite the log in exponential form. To evaluate $\log_5 125$, ask "5 to what power is 125?" Since $5^3 = 125$, the answer is 3.
• Memorize two quick facts: $\log_b 1 = 0$ (because $b^0 = 1$) and $\log_b b = 1$ (because $b^1 = b$).
• On no-calculator sections, expect arguments that are clean powers of the base. If the argument is not a clean power, the answer is likely meant to be estimated or left in log form.

Problem Solving

• When solving $b^x = c$, taking $\log_b$ of both sides isolates $x$ as $x = \log_b c$.
• Check the base and argument restrictions before you start. A log of zero or a negative number is undefined, which can rule out answer choices.
• On a log scale, read distances as ratios. Moving from $10^2$ to $10^4$ is two steps, meaning the value grew by a factor of $10^2 = 100$.

Common Trap

Show the exponential rewrite when work is expected. Writing $\log_2 64 = \log_2 2^6 = 6$ makes your reasoning clear, which is important for clean exam work.

Common Misconceptions

• A log is not multiplication or a fancy exponent by itself. $\log_b c$ is the exponent you put on $b$ to get $c$, so always translate it back to $b^a = c$.
• The argument of a log cannot be zero or negative. There is no real value for $\log_b 0$ or $\log_b(-4)$, because a positive base raised to any real power never gives zero or a negative number.
• $\log c$ with no base is base 10, not base $e$. The natural log base $e$ is written $\ln c$. Mixing these up changes your answer.
• The base cannot be 1 and must be positive. With base 1, every power is 1, so the log would not have a single defined value.
• A logarithmic scale is not linear. Equal spacing on a log scale means equal ratios, so each step multiplies by the base instead of adding a fixed amount.
• $\log_b 1$ is always 0, not 1. Any base to the zero power is 1, so the log of 1 is 0 regardless of the base.

Related AP Precalculus Guides

• 2.8 Inverse Functions
• 2.11 Logarithmic Functions
• 2.4 Exponential Function Manipulation
• 2.7 Composition of Functions
• 2.12 Logarithmic Function Manipulation
• 2.3 Exponential Functions