Logarithmic and Exponential Functions

1. What are Logarithms?

The Relationship Between Logarithms and Indices

A logarithm is simply another way of writing an index equation (also called an exponential equation).

The Basic Relationship:

If a^x = b, then log_a(b) = x

This means: "The power you raise a to, to get b, is x"

Example:

• We know that 2³ = 8
• In logarithm form, this is written as: log₂(8) = 3
• In words: "The power you raise 2 to, to get 8, is 3"

More Examples:

• 10² = 100, so log₁₀(100) = 2
• 5³ = 125, so log₅(125) = 3
• 3⁴ = 81, so log₃(81) = 4

Converting Between Forms:

The Three Main Laws

For any positive number a (where a > 0):

Law 1: Multiplication Rule log_a(xy) = log_a(x) + log_a(y)

This relates to the index law: a^x × a^y = a^(x+y)

Example: log₂(8 × 4) = log₂(8) + log₂(4) = 3 + 2 = 5 (Check: 8 × 4 = 32 = 2⁵, so log₂(32) = 5 ✓)

Law 2: Division Rule log_a(x/y) = log_a(x) - log_a(y)

This relates to the index law: a^x ÷ a^y = a^(x-y)

Example: log₃(27/9) = log₃(27) - log₃(9) = 3 - 2 = 1 (Check: 27/9 = 3 = 3¹, so log₃(3) = 1 ✓)

Law 3: Power Rule log_a(x^k) = k × log_a(x)

This relates to the index law: (a^x)^y = a^(xy)

Example: log₂(16) = log₂(2⁴) = 4 × log₂(2) = 4 × 1 = 4

Important Special Results

These follow from the laws above:

1. log_a(a) = 1

   • Because a¹ = a
   • "The power you raise a to, to get a, is 1"

2. log_a(1) = 0

   • Because a⁰ = 1
   • "The power you raise a to, to get 1, is 0"

3. log_a(a^x) = x

   • Using the power rule: log_a(a^x) = x × log_a(a) = x × 1 = x

4. a^(log_a(x)) = x

   • This shows that raising to a power and taking a log are inverse operations

5. log_a(1/x) = -log_a(x)

   • Because 1/x = x^(-1)
   • So log_a(x^(-1)) = -1 × log_a(x) = -log_a(x)