# Logarithmic expressions

> AP Precalculus · Unit 2: Exponential and Logarithmic Functions
> Source: https://www.owlsprep.com/study/ap-precalculus-u2-logarithmic-expressions/

This guide covers the definition of logarithmic expressions, logarithm-exponential equivalence, core logarithm properties, change of base formula, and techniques for expanding and condensing expressions, a foundational skill for AP Precalculus Unit 2.

**Prerequisites:** [Exponential function definitions and properties](https://www.owlsprep.com/study/ap-precalculus-u2-exponential-functions/); Algebraic rules for exponents; Basic factoring and simplification of algebraic expressions

## Learning objectives

- Define logarithmic expressions and recall their inverse relationship to exponential functions
- Convert between exponential and logarithmic forms using core equivalence
- Apply product, quotient, and power rules to expand and condense logarithmic expressions
- Use the change of base formula to evaluate and simplify logarithms
- Avoid common domain and rule application errors on AP exam questions

## Definition and Logarithm-Exponential Equivalence

A logarithmic expression is any algebraic expression containing one or more logarithms, which are the inverse functions of exponential functions. Logarithms answer the question: what exponent do I raise a given base to, to get the argument value? On the AP exam, you will primarily work with common logarithms (base 10, written $\text{log }x$) and natural logarithms (base $e$, written $\text{ln }x$), though any positive base $b \neq 1$ is valid. Mastery of this topic is required for nearly all subsequent Unit 2 content, so errors here cascade into higher-weight problems.

**Logarithm-Exponential Equivalence** — The entire definition of a logarithm rests on this inverse relationship with exponentiation. It allows free conversion between exponential and logarithmic forms. Domain restrictions require $b>0, b\neq 1, x>0$: negative arguments or invalid bases produce non-real results.

*Notation:* $y = \log_b x \iff b^y = x$

**Worked example:** Evaluate $\log_4 \left(\frac{1}{64}\right)$

1. Let $y = \log_4 \left(\frac{1}{64}\right)$. By equivalence, this translates to:
2. $$4^y = \frac{1}{64}$$
3. Rewrite both sides with base 4 to match the logarithm's base:
4. $$4^y = (4^3)^{-1} = 4^{-3}$$
5. Exponential functions are one-to-one, so exponents are equal, giving $y = -3$. Verify by checking $4^{-3} = \frac{1}{64}$, which matches the original argument.

> **Exam tip:** If you are ever unsure of a logarithm value, convert it back to exponential form to check—this takes 10 seconds and eliminates almost all sign errors.

## Core Logarithm Properties: Expanding and Condensing

Because logarithms are inverses of exponents, their properties directly correspond to familiar exponent rules. For any positive base $b \neq 1$, positive arguments $M, N$, and real number $k$, the three core properties are used for two common AP exam tasks: expanding a single condensed expression into simpler terms, or condensing a sum/difference of logs into a single expression.

1. Product Rule: $\log_b(MN) = \log_b M + \log_b N$ (corresponds to $b^m b^n = b^{m+n}$)
2. Quotient Rule: $\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$ (corresponds to $\frac{b^m}{b^n} = b^{m-n}$)
3. Power Rule: $\log_b(M^k) = k \log_b M$ (corresponds to $(b^m)^k = b^{mk}$)

**Worked example:** Expand $\log_3\left(\frac{9x^5}{\sqrt{y}}\right)$ fully, where $x>0, y>0$.

1. Apply the quotient rule first to split the fraction:
2. $$\log_3\left(\frac{9x^5}{\sqrt{y}}\right) = \log_3(9x^5) - \log_3(\sqrt{y})$$
3. Apply the product rule to the first term, then rewrite the square root as an exponent:
4. $$\log_3 9 + \log_3(x^5) - \log_3(y^{1/2})$$
5. Evaluate the constant log term and apply the power rule to variable terms: $\log_3 9 = 2$, so the fully expanded form is:
6. $$2 + 5\log_3 x - \frac{1}{2}\log_3 y$$

> **Exam tip:** When condensing expressions, always move coefficients inside logarithms as exponents before combining terms with product/quotient rules—this avoids common coefficient errors.

## Change of Base Formula

The change of base formula rewrites a logarithm of any base as a ratio of logarithms with another base. This is required to evaluate any non-common/non-natural logarithm with a calculator, and to simplify expressions with mixed bases.

**Change of Base Formula** — For any positive $a, b \neq 1$, and positive $x$, this formula holds. On the AP exam, use $a=10$ (common log) or $a=e$ (natural log) for calculator evaluation.

*Notation:* $\log_b x = \frac{\log_a x}{\log_a b}$

**Worked example:** Simplify $\frac{\log_2 27}{\log_2 9}$ to a single constant.

1. The expression matches the right-hand side of the change of base formula, with $a=2$, $b=9$, $x=27$. This simplifies directly to:
2. $$\log_9 27$$
3. Rewrite 27 and 9 as powers of the same base 3:
4. $$\log_{3^2} 3^3$$
5. Apply change of base again to rewrite in base 3:
6. $$\frac{\log_3 3^3}{\log_3 3^2} = \frac{3 \log_3 3}{2 \log_3 3}$$
7. Cancel $\log_3 3 = 1$ to get the final result:
8. $$\frac{3}{2}$$

> **Exam tip:** When asked to evaluate a logarithm where both the base and argument are powers of a common smaller base, always rewrite to cancel logs and avoid unnecessary calculator work.

## AP-Style Worked Practice

**Check your understanding**

Test your understanding with this multiple-choice question:

1. Which of the following is equivalent to $2 \log 6 - \log 3$?

   - $\log 3$
   - $\log 8$
   - $\log 12$
   - $\log 36$

   *Answer:* $\log 12$

   *Why:* Correct! You applied the power rule first to move the coefficient, then used the quotient rule to combine the terms.

**Worked example:** Let $x>0, y>0$, and let $A = \log_2(16x^3y^{-2})$. (a) Expand $A$ fully; (b) Calculate $A$ given $\log_2 x = 4$ and $\log_2 y = -1$; (c) Rewrite $A$ in terms of $\ln x$ and $\ln y$.

1. Part (a): Apply product rule to split the argument, then power rule and evaluate the constant log:
2. $$\log_2 16 + \log_2 x^3 + \log_2 y^{-2} = 4 + 3\log_2 x - 2\log_2 y$$
3. Part (b): Substitute the given values:
4. $$A = 4 + 3(4) - 2(-1) = 4 + 12 + 2 = 18$$
5. Part (c): Use change of base to convert base 2 logs to natural logs, $\log_2 x = \frac{\ln x}{\ln 2}$ and $\log_2 y = \frac{\ln y}{\ln 2}$:
6. $$A = 4 + \frac{3\ln x - 2\ln y}{\ln 2}$$

**Worked example:** The pH of a solution is given by $\text{pH} = -\log_{10} [\text{H}^+]$, where $[\text{H}^+]$ is hydrogen ion concentration in mol/L. Orange juice has $[\text{H}^+] = 3.2 \times 10^{-4}$ mol/L. Calculate its pH to one decimal place. Milk has pH 6.6, find its $[\text{H}^+]$ in scientific notation.

1. For orange juice, substitute into the formula and expand using product rule:
2. $$\text{pH} = -\log_{10}(3.2 \times 10^{-4}) = -\left(\log_{10} 3.2 + \log_{10} 10^{-4}\right) = -(0.505 - 4) \approx 3.5$$
3. For milk, rearrange and convert to exponential form:
4. $$6.6 = -\log_{10} [\text{H}^+] \implies [\text{H}^+] = 10^{-6.6} = 10^{0.4} \times 10^{-7} \approx 2.5 \times 10^{-7}$$

## Common pitfalls

- **Wrong:** $\log_b(M+N) = \log_b M + \log_b N$
  - Why it fails: Students confuse the product rule for logs with addition inside the log; no general rule exists for the logarithm of a sum.
  - Correct: Only apply logarithm rules to product, quotient, and power operations inside the log, never addition or subtraction.
- **Wrong:** $\frac{\log_b M}{\log_b N} = \log_b\left(\frac{M}{N}\right)$
  - Why it fails: Students confuse the change of base formula (quotient of two separate logs) with the quotient rule (which applies to a quotient inside a single log).
  - Correct: Only apply the quotient rule when division is inside the logarithm, not between two separate logarithms.
- **Wrong:** $\log_b(-x) = -\log_b x$ for $x>0$
  - Why it fails: Students incorrectly apply the power rule to a negative argument, forgetting the domain requirement that all logarithm arguments must be positive.
  - Correct: Always check domain first: $\log_b(-x)$ is only defined for $x<0$, and a negative sign in the argument can never be pulled out as a coefficient.
- **Wrong:** $(\log_b M)^k = k \log_b M$
  - Why it fails: Students misapply the power rule, which only applies to a power inside the logarithm, not to the entire logarithm raised to a power.
  - Correct: Only use the power rule when the exponent is on the argument of the log, not on the log itself.
- **Wrong:** $\log_b\left(\frac{1}{x}\right) = \frac{1}{\log_b x}$
  - Why it fails: Students incorrectly apply reciprocal rules from general algebra to logarithms, forgetting the power rule for reciprocals.
  - Correct: Rewrite $\frac{1}{x}$ as $x^{-1}$, then apply the power rule to get $\log_b\left(\frac{1}{x}\right) = -\log_b x$.

## Cheatsheet

| Category | Formula | Notes |
| --- | --- | --- |
| Core Definition | $y = \log_b x \iff b^y = x$ | $b>0, b\neq 1, x>0$ |
| Common Logarithm | $\log x = \log_{10} x$ | No base written explicitly; used for pH, Richter scale |
| Natural Logarithm | $\ln x = \log_e x$ | Base $e \approx 2.71828$; used for continuous growth |
| Product Rule | $\log_b(MN) = \log_b M + \log_b N$ | $M>0, N>0$; works for any number of factors |
| Quotient Rule | $\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$ | $M>0, N>0$ |
| Power Rule | $\log_b(M^k) = k \log_b M$ | $M>0$; works for any real $k$ |
| Reciprocal Rule | $\log_b\left(\frac{1}{M}\right) = -\log_b M$ | Special case of power rule with $k=-1$ |
| Change of Base Formula | $\log_b x = \frac{\log_a x}{\log_a b}$ | Use $a=10$ or $a=e$ for calculator evaluation |
| Logarithm of Base | $\log_b b = 1$ | Follows directly from core definition |
| Logarithm of 1 | $\log_b 1 = 0$ | Follows directly from core definition |

## What's next

Mastery of logarithmic expressions is the foundational prerequisite for all remaining topics in AP Precalculus Unit 2: Exponential and Logarithmic Functions, which makes up 25-30% of your total AP exam score. You will apply these simplification and manipulation techniques to isolate variables when solving equations, model real-world growth and decay, and graph logarithmic functions. Errors in simplifying logarithmic expressions almost always lead to incorrect answers on higher-weight problems, so be sure to master this topic before moving on.

- [Inverses of exponential functions](https://www.owlsprep.com/study/ap-precalculus-u2-inverses-of-exponential-functions/)
- [Logarithmic Functions](https://www.owlsprep.com/study/ap-precalculus-u2-logarithmic-functions/)
- [Logarithmic function manipulation](https://www.owlsprep.com/study/ap-precalculus-u2-logarithmic-function-manipulation/)

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