Logarithms

What is a Logarithm?

Logarithm: The inverse operation of exponentiation

If b^x = y, then log_b(y) = x

In words: "The logarithm base b of y equals x"

Read as: "log base b of y"

Connection to Exponents

Exponential form: 2³ = 8

Logarithmic form: log₂(8) = 3

Both say: "2 raised to what power equals 8? Answer: 3"

Converting Between Forms

Exponential to Logarithmic:

• b^x = y → log_b(y) = x

Logarithmic to Exponential:

• log_b(y) = x → b^x = y

Example 1: Convert to Log Form

Exponential: 5² = 25

Logarithmic: log₅(25) = 2

Meaning: "5 to what power equals 25? 2"

Example 2: Convert to Exponential

Logarithmic: log₃(81) = 4

Exponential: 3⁴ = 81

Example 3: With Variables

Exponential: 10^x = 1000

Logarithmic: log₁₀(1000) = x

Solve: x = 3 (since 10³ = 1000)

Common Logarithms

Common log: Base 10, written as log(x) or log₁₀(x)

On calculator: LOG button

Examples:

• log(10) = 1 (because 10¹ = 10)
• log(100) = 2 (because 10² = 100)
• log(1000) = 3 (because 10³ = 1000)

Example: Common Log

Evaluate: log(10,000)

Think: 10 to what power = 10,000?

Answer: 10⁴ = 10,000, so log(10,000) = 4

Natural Logarithms

Natural log: Base e ≈ 2.718, written as ln(x)

On calculator: LN button

Definition: If e^x = y, then ln(y) = x

Examples:

• ln(e) = 1 (because e¹ = e)
• ln(1) = 0 (because e⁰ = 1)

Evaluating Logarithms

Find the value of log expressions

Example 1: Simple Log

Evaluate: log₂(16)

Ask: 2 to what power = 16?

2¹ = 2
2² = 4
2³ = 8
2⁴ = 16

Answer: log₂(16) = 4

Example 2: Fraction Result

Evaluate: log₄(2)

Ask: 4 to what power = 2?

4^(1/2) = √4 = 2

Answer: log₄(2) = 1/2

Example 3: Negative Exponent

Evaluate: log₅(1/25)

Ask: 5 to what power = 1/25?

1/25 = 1/5² = 5^(-2)

Answer: log₅(1/25) = -2

Special Log Values

For any base b > 0, b ≠ 1:

log_b(1) = 0

• Because b⁰ = 1

log_b(b) = 1

• Because b¹ = b

log_b(b^x) = x

• Because b^x = b^x

Examples

• log₇(1) = 0
• log₇(7) = 1
• log₇(7³) = 3

Properties of Logarithms

Product Property: log_b(MN) = log_b(M) + log_b(N)

Quotient Property: log_b(M/N) = log_b(M) - log_b(N)

Power Property: log_b(M^p) = p · log_b(M)

Example 1: Product Property

Simplify: log₂(8) + log₂(4)

Using property:

log₂(8 · 4) = log₂(32)

Evaluate:

2⁵ = 32, so log₂(32) = 5

Answer: 5

Example 2: Quotient Property

Simplify: log₃(27) - log₃(3)

Using property:

log₃(27/3) = log₃(9)

Evaluate:

3² = 9, so log₃(9) = 2

Answer: 2

Example 3: Power Property

Simplify: log₅(25²)

Using property:

2 · log₅(25)

Evaluate:

log₅(25) = 2 (since 5² = 25)
2 · 2 = 4

Answer: 4

Expanding Logarithms

Use properties to write as sum/difference

Example 1: Expand

Expand: log₂(8x)

Product property:

log₂(8) + log₂(x)
= 3 + log₂(x)

Example 2: Complex Expansion

Expand: log(x³/100)

Apply quotient then power:

log(x³) - log(100)
= 3log(x) - 2

Condensing Logarithms

Combine into single log

Example 1: Condense

Condense: log₃(5) + log₃(9)

Product property:

log₃(5 · 9) = log₃(45)

Example 2: With Coefficients

Condense: 2log(x) - log(y)

Apply power then quotient:

log(x²) - log(y) = log(x²/y)

Solving Logarithmic Equations

Isolate the log, then convert to exponential form

Example 1: Simple Log Equation

Solve: log₂(x) = 5

Convert to exponential:

2⁵ = x
x = 32

Answer: x = 32

Example 2: With Addition

Solve: log₃(x) + 2 = 5

Isolate log:

log₃(x) = 3

Convert:

3³ = x
x = 27

Answer: x = 27

Example 3: Use Properties

Solve: log₄(x) + log₄(3) = 2

Product property:

log₄(3x) = 2

Convert:

4² = 3x
16 = 3x
x = 16/3

Answer: x = 16/3

Solving Exponential Equations with Logs

Use logs to solve for variable in exponent

Example 1: Common Log

Solve: 10^x = 500

Take log of both sides:

log(10^x) = log(500)
x · log(10) = log(500)
x · 1 = log(500)
x = log(500) ≈ 2.699

Example 2: Different Base

Solve: 2^x = 50

Take log of both sides:

log(2^x) = log(50)
x · log(2) = log(50)
x = log(50)/log(2)
x ≈ 5.644

Change of Base Formula

Convert any base to common or natural log

Formula: log_b(x) = log(x)/log(b) or ln(x)/ln(b)

Example: Calculate Log₃(20)

Calculator doesn't have base 3:

Use change of base:

log₃(20) = log(20)/log(3)
         ≈ 1.301/0.477
         ≈ 2.727

Real-World Applications

pH scale: pH = -log[H⁺] (chemistry)

Richter scale: Earthquake magnitude (geology)

Decibels: Sound intensity (physics)

Half-life: Radioactive decay (nuclear science)

Population growth: Exponential models

Example: pH Calculation

If [H⁺] = 10^(-5), find pH

Formula:

pH = -log(10^(-5))
   = -(-5)
   = 5

Answer: pH = 5 (acidic)

Practice