📈 Logarithmic Functions

Part 1 of 7 — Logarithm Basics

1. log_b(x) = y means b^y = x

log_b(x) = y means b^y = x

2. Logarithm is the inverse of the exponential function

Logarithm is the inverse of the exponential function

3. Domain

(0, ∞); Range: all real numbers

4. The graph of y = log_b(x) passes through (1, 0) and (b, 1)

The graph of y = log_b(x) passes through (1, 0) and (b, 1)

Check Your Understanding 🎯

Key Concepts Summary

• log_b(x) = y means b^y = x
• Logarithm is the inverse of the exponential function
• Domain: (0, ∞); Range: all real numbers
• The graph of y = log_b(x) passes through (1, 0) and (b, 1)

Concept Check 🎯

Match the Concepts 🔍

Properties of Logarithms

Part 2 of 7 — Properties of Logarithms

1. Product rule

log_b(MN) = log_b(M) + log_b(N)

2. Quotient rule

log_b(M/N) = log_b(M) - log_b(N)

3. Power rule

log_b(M^n) = n · log_b(M)

4. Change of base

log_b(x) = ln(x)/ln(b) = log(x)/log(b)

Check Your Understanding 🎯

Key Concepts Summary

• Product rule: log_b(MN) = log_b(M) + log_b(N)
• Quotient rule: log_b(M/N) = log_b(M) - log_b(N)
• Power rule: log_b(M^n) = n · log_b(M)
• Change of base: log_b(x) = ln(x)/ln(b) = log(x)/log(b)

Concept Check 🎯

Match the Concepts 🔍

Common & Natural Logs

Part 3 of 7 — Common & Natural Logs

1. Common log

log(x) = log₁₀(x), used for pH, decibels, Richter scale

2. Natural log

ln(x) = logₑ(x), used in calculus and natural phenomena

3. ln(e) = 1 and log(10) = 1

ln(e) = 1 and log(10) = 1

4. ln(eˣ) = x and e^(ln(x)) = x for x > 0

ln(eˣ) = x and e^(ln(x)) = x for x > 0

Check Your Understanding 🎯

Key Concepts Summary

• Common log: log(x) = log₁₀(x), used for pH, decibels, Richter scale
• Natural log: ln(x) = logₑ(x), used in calculus and natural phenomena
• ln(e) = 1 and log(10) = 1
• ln(eˣ) = x and e^(ln(x)) = x for x > 0

Concept Check 🎯

Match the Concepts 🔍

Solving Logarithmic Equations

Part 4 of 7 — Solving Logarithmic Equations

1. Isolate the logarithmic expression

Isolate the logarithmic expression

2. Convert to exponential form

log_b(x) = y → b^y = x

3. Check for extraneous solutions (argument must be positive)

Check for extraneous solutions (argument must be positive)

4. Use properties to combine or expand log expressions before solving

Use properties to combine or expand log expressions before solving

Check Your Understanding 🎯

Key Concepts Summary

• Isolate the logarithmic expression
• Convert to exponential form: log_b(x) = y → b^y = x
• Check for extraneous solutions (argument must be positive)
• Use properties to combine or expand log expressions before solving

Concept Check 🎯

Match the Concepts 🔍

Logarithmic Modeling

Part 5 of 7 — Logarithmic Modeling

1. Richter scale

M = log(I/I₀), each unit is 10× intensity

2. Decibel scale

dB = 10 · log(I/I₀)

3. pH scale

pH = -log[H⁺], logarithmic measure of acidity

4. Logarithmic regression

y = a + b · ln(x) for data modeling

Check Your Understanding 🎯

Key Concepts Summary

• Richter scale: M = log(I/I₀), each unit is 10× intensity
• Decibel scale: dB = 10 · log(I/I₀)
• pH scale: pH = -log[H⁺], logarithmic measure of acidity
• Logarithmic regression: y = a + b · ln(x) for data modeling

Concept Check 🎯

Match the Concepts 🔍

Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

1. Richter scale

M = log(I/I₀), each unit is 10× intensity

2. Decibel scale

dB = 10 · log(I/I₀)

3. pH scale

pH = -log[H⁺], logarithmic measure of acidity

4. Logarithmic regression

y = a + b · ln(x) for data modeling

Check Your Understanding 🎯

Key Concepts Summary

• Richter scale: M = log(I/I₀), each unit is 10× intensity
• Decibel scale: dB = 10 · log(I/I₀)
• pH scale: pH = -log[H⁺], logarithmic measure of acidity
• Logarithmic regression: y = a + b · ln(x) for data modeling

Concept Check 🎯

Match the Concepts 🔍

Review & Applications

Part 7 of 7 — Review & Applications

1. Richter scale

M = log(I/I₀), each unit is 10× intensity

2. Decibel scale

dB = 10 · log(I/I₀)

3. pH scale

pH = -log[H⁺], logarithmic measure of acidity

4. Logarithmic regression

y = a + b · ln(x) for data modeling

Check Your Understanding 🎯

Key Concepts Summary

• Richter scale: M = log(I/I₀), each unit is 10× intensity
• Decibel scale: dB = 10 · log(I/I₀)
• pH scale: pH = -log[H⁺], logarithmic measure of acidity
• Logarithmic regression: y = a + b · ln(x) for data modeling

Concept Check 🎯

Match the Concepts 🔍