This lesson introduces the fundamental concept of logarithms. A logarithm is simply another way to write an index (or a power). Mastering the connection between index form and log form is the key to this topic.

1. The Core Relationship: Index Form vs. Log Form

This is the most important concept to memorize. Every log problem can be understood by relating it back to indices.

The Rule: aˣ = N <=> logₐ(N) = x

• Index Form (aˣ = N): "The base a raised to the power x equals the number N."

  • Example: 2³ = 8

• Logarithm Form (logₐ(N) = x): "The power you must raise the base a to, in order to get the number N, is x."

  • Example: log₂(8) = 3

How to Convert:

• Index to Log: 10² = 100 → log₁₀(100) = 2. (The base of the power becomes the base of the log).

• Log to Index: log₅(25) = 2 → 5² = 25. (The base of the log "swings" over to become the base for the power).

2. The Two Laws of Logarithms

These laws come directly from the laws of indices and are used to simplify expressions.

1. The Product Law (for Multiplication): When you add logs, you multiply their numbers. logₐ(m) + logₐ(n) = logₐ(m × n)

2. The Quotient Law (for Division): When you subtract logs, you divide their numbers. logₐ(m) - logₐ(n) = logₐ(m / n)

Two Special Cases (Shortcuts):

• logₐ(a) = 1 (because a¹ = a)

• logₐ(1) = 0 (because a⁰ = 1)

3. How to Solve Exam Questions

Type 1: "Evaluate" or "Find the Value"

• Question: Find the value of log₃(54) - log₃(2).

• Steps:

  1. Combine using a log law: The law for subtraction is division. log₃(54 / 2) = log₃(27)

  2. Convert to index thinking: Ask yourself: "What power of 3 gives 27?" 3 × 3 × 3 = 27, so 3³ = 27.

  3. Answer: The value is 3.

Type 2: "Solve for x"

• Question: Solve log₅(x) + log₅(4) = log₅(20).

• Steps:

  1. Combine logs into a single log on each side: Use the log laws. log₅(x × 4) = log₅(20) log₅(4x) = log₅(20)

  2. Equate the numbers: Since the logs and their bases are the same, the numbers inside must be equal. 4x = 20

  3. Solve: x = 5.

Type 3: "Express in terms of..."

• Question: If logₐ(2) = p and logₐ(3) = q, express logₐ(18) in terms of p and q.

• Steps:

  1. Break down the number: Write 18 using only the numbers you have logs for (2 and 3). 18 = 2 × 9 = 2 × 3 × 3

  2. Rewrite the log: logₐ(18) = logₐ(2 × 3 × 3)

  3. Expand using the log laws: Multiplication becomes addition. logₐ(2) + logₐ(3) + logₐ(3)

  4. Substitute the given letters: p + q + q = p + 2q.

4. Important Points to Remember

• The "Same Base" Rule: You can only use the log laws to combine terms that have the exact same base.

• A log is a power. If you ever get stuck, convert the log back to the more familiar index form to understand what it means.

• You can only take the log of a positive number. log₂(-8) is not defined.