Logarithms

Overview

Logarithms answer exponent questions in reverse. If $2^3 = 8$, then $\log_2 8 = 3$. The logarithm tells the exponent needed to produce a number from a chosen base.

This topic matters because it joins several Form II algebra skills: powers, roots, equations, standard form, and careful use of rules. Logarithms are also useful when a calculation involves very large numbers, very small numbers, or products that are easier to handle by breaking them into factors.

Prerequisites

• Exponents - Logarithms reverse exponentiation, so exponent notation and laws must be secure first.
• Radicals - Some logarithm expressions include roots, such as $\log_a \sqrt{a}$.
• Standard form - Common logarithms often appear with powers of $10$.
• Algebraic expressions and equations - Logarithmic equations require substitution, simplification, and checking restrictions.
• Rational, irrational, and real numbers - Logarithm inputs and bases must be handled as real positive quantities in this chapter.

Learning Scope

This chapter covers the meaning of logarithms, conversion between exponential and logarithmic form, basic logarithm laws, simple evaluation, common logarithms, and simple logarithmic equations.

This page does not fully teach advanced logarithmic functions, graphs of logarithmic functions, natural logarithms in calculus, or numerical methods. It also does not present official past-paper solutions; exam-derived items are listed only as unreviewed signals for later review.

Subtopics

Meaning Of A Logarithm

A logarithm asks: "What exponent gives this number?"

If:

$$ a^x = N $$

then:

$$ \log_a N = x $$

The base is $a$, the number being logged is $N$, and the answer is the exponent $x$.

Example:

$$ 3^4 = 81 $$

so:

$$ \log_3 81 = 4 $$

Key insight: A logarithm is not a new kind of multiplication. It is an exponent written from the reverse direction.

Conditions For Logarithms

For real logarithms in this chapter:

$$ \log_a N $$

is defined when:

$$ a > 0, \quad a \ne 1, \quad N > 0 $$

The base must be positive and not equal to $1$. The number inside the logarithm must also be positive.

Key insight: Before solving a logarithmic equation, remember that every logarithm input must be greater than $0$.

Converting Between Forms

Logarithmic and exponential forms say the same fact in two different ways:

$$ \log_a N = x \quad \Longleftrightarrow \quad a^x = N $$

Examples:

$$ \log_2 32 = 5 \quad \Longleftrightarrow \quad 2^5 = 32 $$

$$ \log_{10} 1000 = 3 \quad \Longleftrightarrow \quad 10^3 = 1000 $$

Converting forms is often the fastest way to evaluate a simple logarithm.

Basic Values

Some logarithm values follow directly from exponent facts:

$$ \log_a a = 1 $$

because:

$$ a^1 = a $$

Also:

$$ \log_a 1 = 0 $$

because:

$$ a^0 = 1 $$

Examples:

$$ \log_5 5 = 1 $$

$$ \log_7 1 = 0 $$

Product Law

The logarithm of a product can be split into a sum:

$$ \log_a (MN) = \log_a M + \log_a N $$

where $a > 0$, $a \ne 1$, $M > 0$, and $N > 0$.

Example:

$$ \log_2 (8 \times 4) = \log_2 8 + \log_2 4 = 3 + 2 = 5 $$

This agrees with:

$$ \log_2 32 = 5 $$

Key insight: Multiplication inside a logarithm becomes addition outside the logarithm.

Quotient Law

The logarithm of a quotient can be split into a difference:

$$ \log_a \left(\frac{M}{N}\right) = \log_a M - \log_a N $$

where $a > 0$, $a \ne 1$, $M > 0$, and $N > 0$.

Example:

$$ \log_3 \left(\frac{81}{9}\right) = \log_3 81 - \log_3 9 = 4 - 2 = 2 $$

This agrees with:

$$ \log_3 9 = 2 $$

Power Law

An exponent inside a logarithm can be brought to the front:

$$ \log_a (M^n) = n\log_a M $$

Example:

$$ \log_2 (8^3) = 3\log_2 8 = 3 \times 3 = 9 $$

This agrees with:

$$ 8^3 = (2^3)^3 = 2^9 $$

so:

$$ \log_2 (8^3) = 9 $$

Logarithms With Roots

Roots can be rewritten using fractional exponents:

$$ \sqrt{a} = a^{\frac{1}{2}} $$

So:

$$ \log_a \sqrt{a} = \log_a \left(a^{\frac{1}{2}}\right) = \frac{1}{2}\log_a a = \frac{1}{2} $$

Key insight: A root inside a logarithm often becomes easier after rewriting the root as a power.

Common Logarithms

When no base is written, school algebra often treats $\log N$ as a common logarithm:

$$ \log N = \log_{10} N $$

Common logarithms use base $10$:

$$ \log 100 = 2 $$

because:

$$ 10^2 = 100 $$

Given values such as $\log 2 = 0.3010$ and $\log 3 = 0.4771$ can be combined with logarithm laws to find other logarithms.

Solving Simple Logarithmic Equations

A logarithmic equation may be solved by using logarithm laws, converting to exponential form, or both.

Example idea:

$$ \log_2 x = 5 $$

Convert to exponential form:

$$ 2^5 = x $$

So:

$$ x = 32 $$

For equations with expressions inside logarithms, solve algebraically and then check that every logarithm input is positive.

Checking Restrictions

Consider an equation containing $\log_{10}(y - 4)$. This expression is defined only when:

$$ y - 4 > 0 $$

so:

$$ y > 4 $$

If algebra gives $y = 3$, that value must be rejected because $\log_{10}(3 - 4)$ is not a real logarithm.

Key insight: Solving the equation is not enough; the answer must also satisfy the logarithm restrictions.

Key Terms

• Logarithm: The exponent needed to produce a number from a given base.
• Base: The number being raised to a power in the matching exponential form. In $\log_2 8$, the base is $2$.
• Argument: The number or expression inside the logarithm. In $\log_2 8$, the argument is $8$.
• Common logarithm: A logarithm with base $10$, often written as $\log N$.
• Product law: The rule $\log_a(MN) = \log_a M + \log_a N$.
• Quotient law: The rule $\log_a\left(\frac{M}{N}\right) = \log_a M - \log_a N$.
• Power law: The rule $\log_a(M^n) = n\log_a M$.
• Domain restriction: A condition such as $N > 0$ that must be true before $\log_a N$ is defined in real-number work.

Worked Examples

Example 1: Convert To Logarithmic Form

Write $5^3 = 125$ in logarithmic form.

The exponent is $3$, the base is $5$, and the result is $125$:

$$ 5^3 = 125 $$

Therefore:

$$ \log_5 125 = 3 $$

Final answer:

$$ \log_5 125 = 3 $$

Example 2: Evaluate A Logarithm

Evaluate $\log_4 64$.

Ask which power of $4$ gives $64$:

$$ 4^1 = 4 $$

$$ 4^2 = 16 $$

$$ 4^3 = 64 $$

Therefore:

$$ \log_4 64 = 3 $$

Final answer:

$$ 3 $$

Example 3: Simplify Using Logarithm Laws

Simplify:

$$ \log_a \sqrt{a} + \log_a(a^2) $$

Rewrite the root as a fractional power:

$$ \sqrt{a} = a^{\frac{1}{2}} $$

Then:

$$ \begin{aligned} \log_a \sqrt{a} + \log_a(a^2) &= \log_a\left(a^{\frac{1}{2}}\right) + \log_a(a^2) \\ &= \frac{1}{2}\log_a a + 2\log_a a \\ &= \frac{1}{2} + 2 \\ &= \frac{5}{2} \end{aligned} $$

Final answer:

$$ \frac{5}{2} $$

Example 4: Use Given Common Logarithms

Given $\log 2 = 0.3010$ and $\log 3 = 0.4771$, find $\log 72$.

Factor $72$ using $2$ and $3$:

$$ 72 = 8 \times 9 = 2^3 \times 3^2 $$

Use the product and power laws:

$$ \begin{aligned} \log 72 &= \log(2^3 \times 3^2) \\ &= \log(2^3) + \log(3^2) \\ &= 3\log 2 + 2\log 3 \\ &= 3(0.3010) + 2(0.4771) \\ &= 0.9030 + 0.9542 \\ &= 1.8572 \end{aligned} $$

Final answer:

$$ \log 72 = 1.8572 $$

Example 5: Solve A Logarithmic Equation

Solve:

$$ 4 + 3\log_3 x = \log_3 24 $$

First write $4$ as a logarithm with base $3$:

$$ 4 = \log_3(3^4) = \log_3 81 $$

Then use the power law:

$$ 3\log_3 x = \log_3(x^3) $$

Now combine:

$$ \begin{aligned} 4 + 3\log_3 x &= \log_3 24 \\ \log_3 81 + \log_3(x^3) &= \log_3 24 \\ \log_3(81x^3) &= \log_3 24 \end{aligned} $$

Since the logarithms have the same valid base:

$$ 81x^3 = 24 $$

So:

$$ \begin{aligned} x^3 &= \frac{24}{81} \\ &= \frac{8}{27} \\ x &= \frac{2}{3} \end{aligned} $$

Check the restriction:

$$ x > 0 $$

and $\frac{2}{3} > 0$, so the solution is valid.

Final answer:

$$ x = \frac{2}{3} $$

Example 6: Solve And Check The Domain

Solve:

$$ \log_{10}(3y + 2) - 1 = \log_{10}(y - 4) $$

Write $1$ as $\log_{10} 10$:

$$ 1 = \log_{10} 10 $$

Then:

$$ \begin{aligned} \log_{10}(3y + 2) - \log_{10} 10 &= \log_{10}(y - 4) \\ \log_{10}\left(\frac{3y + 2}{10}\right) &= \log_{10}(y - 4) \end{aligned} $$

Equate the arguments:

$$ \frac{3y + 2}{10} = y - 4 $$

Solve:

$$ \begin{aligned} 3y + 2 &= 10y - 40 \\ 42 &= 7y \\ y &= 6 \end{aligned} $$

Check restrictions:

$$ 3y + 2 > 0 $$

and:

$$ y - 4 > 0 $$

For $y = 6$, both arguments are positive:

$$ 3(6) + 2 = 20 > 0 $$

$$ 6 - 4 = 2 > 0 $$

Final answer:

$$ y = 6 $$

Common Mistakes

• Mistake: Thinking $\log_2 8$ means $2 \times 8$.

Correction: $\log_2 8$ asks for the exponent $x$ such that $2^x = 8$, so $\log_2 8 = 3$.

• Mistake: Forgetting the base when converting forms.

Correction: $\log_a N = x$ means $a^x = N$.

• Mistake: Writing $\log_a(M + N) = \log_a M + \log_a N$.

Correction: The product law works for multiplication, not addition: $\log_a(MN) = \log_a M + \log_a N$.

• Mistake: Applying logarithm laws when arguments are not positive.

Correction: Check that every logarithm argument is greater than $0$.

• Mistake: Treating $\log_a 1$ as $1$.

Correction: $\log_a 1 = 0$ because $a^0 = 1$.

• Mistake: Moving an exponent incorrectly in $\log_a(M^n)$.

Correction: $\log_a(M^n) = n\log_a M$.

• Mistake: Accepting every algebraic solution of a logarithmic equation.

Correction: Substitute the solution into the original logarithm arguments to check validity.

Practice Tasks

Direct Understanding

1. Write $2^6 = 64$ in logarithmic form.
2. Write $\log_5 25 = 2$ in exponential form.
3. State the restrictions on $a$ and $N$ for $\log_a N$ in real-number work.
4. Explain why $\log_7 1 = 0$.

Skill Practice

1. Evaluate $\log_3 27$.
2. Evaluate $\log_2 \frac{1}{8}$.
3. Simplify $\log_a a^5$.
4. Simplify $\log_a \sqrt{a}$.
5. Simplify $\log_4 16 + \log_4 4$.
6. Simplify $\log_5 125 - \log_5 5$.

Application Problems

1. Given $\log 2 = 0.3010$ and $\log 5 = 0.6990$, find $\log 40$.
2. Given $\log 3 = 0.4771$ and $\log 7 = 0.8451$, find $\log 63$.
3. Solve $\log_2 x = 4$.
4. Solve $\log_3(x + 1) = 2$.
5. Solve $\log_{10}(2x - 1) = \log_{10}(x + 3)$.

Multi-Step Reasoning

1. Solve $2 + \log_2 x = \log_2 20$.
2. Solve $\log_5(x - 1) + \log_5 2 = \log_5 18$.
3. Simplify $\log_a(a^3\sqrt{a})$.
4. If $\log_b 81 = 4$, find $b$.
5. Decide whether $x = -2$ can be a solution of an equation containing $\log_3(x + 1)$. Give a reason.

Edge Cases

1. Explain why $\log_1 5$ is not allowed in this chapter.
2. Explain why $\log_2(-8)$ is not a real logarithm in this chapter.
3. Explain why $\log_a(M + N)$ cannot usually be split into two logarithms.
4. Find the mistake: $\log_3 27 = 9$ because $3 \times 9 = 27$.

Generated Question Layer

• Conceptual questions: Ask learners to describe logarithms as exponents and convert between exponential and logarithmic form.
• Skill questions: Generate direct evaluations such as $\log_2 32$, $\log_5 1$, and $\log_{10} 1000$.
• Law-based simplification: Generate product, quotient, and power-law tasks with numeric and algebraic bases.
• Root connection: Generate tasks involving $\log_a \sqrt{a}$, $\log_a \sqrt[3]{a}$, and powers with fractional exponents.
• Equation solving: Generate simple logarithmic equations that require domain checks after solving.
• Common-log practice: Generate tasks using supplied values such as $\log 2$, $\log 3$, and $\log 5$ to find logarithms of composite numbers.
• Misconception checks: Include invalid bases, negative arguments, and false expansions such as $\log_a(M + N)$.

Exam-Derived Signals

The raw 2021-2025 automatic frequency file counts 9 primary Logarithms records. That count is unreviewed and appears to include noisy matches where non-logarithm questions were mapped to Logarithms. Treat it as a pipeline signal, not as an audited topic frequency.

The 2022 examination-format crosswalk lists one combined group, Exponents/Radicals/Logarithms, mapped to topic-exponents, topic-[[radical|radicals]], and topic-logarithms. That is assessment guidance, not a replacement for the syllabus sequence.

Cleaner unreviewed signals from the extracted question map include:

| Year | Question ID | Signal | | ---: | --- | --- | | 2021 | csee_041_2021_p1_q02_b | Solving a base-$10$ logarithmic equation with linear expressions inside logarithms. | | 2022 | csee_041_2022_p1_q02_b_i | Simplifying logarithms involving $\sqrt{a}$ and $a^2$. | | 2023 | csee_041_2023_p1_q02_b | Solving an equation involving $\log_3 x$ and $\log_3 24$. | | 2024 | csee_041_2024_p1_q02_b | Solving a logarithmic equation without using a table or calculator. | | 2025 | csee_041_2025_p1_q02_c | Using given common logarithm values to find $\log 72$. |

These records should be checked against the original papers before they are used as reviewed past-question references.

Source And Review Notes

• Official syllabus status: The topic title, form placement, competence, and hub come from data/curriculum_map.json, which cites raw/syllabuses/csee/2023/csee_mathematics_syllabus_2023.pdf.
• Learner content status: The explanations, examples, and practice tasks are original learner-facing prose drafted from the syllabus topic and standard logarithm concepts.
• Exam signal status: data/question_map_2021_2025.jsonl and data/topic_frequency_2021_2025.json are unreviewed extraction outputs. They are useful for calibration but may contain false positives.
• Exam format status: data/exam_format_topic_crosswalk_2022.jsonl maps the official format group Exponents/Radicals/Logarithms to this topic group, but it does not define the full teaching scope.
• Review risk: Domain restrictions, logarithm-law conditions, and noisy exam mappings should receive manual review before this page is marked reviewed.
• Renderer QA: This page uses $...$ and $$...$$ math notation for compatibility with Obsidian, KaTeX, and MathJax.