Properties of Logarithms - Mathematics

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📐 ∑

Properties of Logarithms

Driving Question

"How can the properties of logarithms help us solve problems faster and more accurately?"

By understanding logarithm properties, we can transform complex forms into simpler ones.

Learning Objectives:

State the 7 basic properties of logarithms.
Apply logarithm properties to simplify and solve problems.
Change the base of logarithms using the change-of-base property.
Determine logarithm values without a calculator.
Use CER (Claim-Evidence-Reasoning) to justify mathematical solutions.

Created by Pak Syah — Mathematics Teacher at SMAN 10 Samarinda

What is a Logarithm?

❓

Logarithm is the inverse operation of exponentiation.

If \( a^b = c \), then \( {}^a \log c = b \)

with conditions:

\( a > 0 \) and \( a \neq 1 \) (base),
\( b > 0 \) (numerus).

\( 2^3 = 8 \) ↔ \( {}^2 \log 8 = 3 \)

Key idea: A logarithm answers the question "What is the exponent?"

Write the exponential form as logarithmic form:

1. \( 5^4 = 625 \) →

✏️ Your Answer:

2. \( 10^3 = 1000 \) →

✏️ Your Answer:

3. \( 2^6 = 64 \) →

✏️ Your Answer:

Created by Pak Syah — SMAN 10 Samarinda

7 Basic Properties

For \( a > 0, a \neq 1, b > 0, c > 0 \)

(i) \( {}^a \log 1 = 0 \)

(ii) \( {}^a \log a = 1 \)

(iii) \( {}^a \log (b \cdot c) = {}^a \log b + {}^a \log c \)

(iv) \( {}^a \log \frac{b}{c} = {}^a \log b - {}^a \log c \)

(v) \( a^{{}^a \log b} = b \)

(vi) \( {}^{a^m} \log b^n = \frac{n}{m} \cdot {}^a \log b \)

(vii) \( {}^a \log b = \frac{{}^p \log b}{{}^p \log a} \), with \( p > 0, p \neq 1 \)

Consequences of property (vii):
\( {}^a \log b = \frac{1}{{}^b \log a} \) | \( {}^a \log b \cdot {}^b \log c = {}^a \log c \)

Created by Pak Syah — SMAN 10 Samarinda

Properties (i) & (ii)

Property (i) — \( {}^a \log 1 = 0 \)

Why? Because \( a^0 = 1 \) for any \( a \neq 0 \).

Example: \( {}^5 \log 1 = 0 \)

Property (ii) — \( {}^a \log a = 1 \)

Why? Because \( a^1 = a \).

Example: \( {}^7 \log 7 = 1 \)

Evaluate the following:

1. \( {}^3 \log 1 = \)

✏️ Your Answer:

2. \( {}^{10} \log 10 = \)

✏️ Your Answer:

3. \( {}^4 \log 1 + {}^9 \log 9 = \)

✏️ Your Answer:

4. \( {}^2 \log 2 \cdot {}^5 \log 1 = \)

✏️ Your Answer:

Created by Pak Syah — SMAN 10 Samarinda

Properties (iii) & (iv)

Property (iii) — Product Rule

\( {}^a \log (b \cdot c) = {}^a \log b + {}^a \log c \)

Example: \( \log 6 = \log (2 \cdot 3) = \log 2 + \log 3 \)

Property (iv) — Quotient Rule

\( {}^a \log \frac{b}{c} = {}^a \log b - {}^a \log c \)

Example: \( \log 5 = \log \frac{10}{2} = \log 10 - \log 2 \)

Given \( \log 2 = 0.301 \), \( \log 3 = 0.477 \), and \( \log 7 = 0.845 \), find:

1. \( \log 6 = \log (2 \cdot 3) = \)

✏️ Your Answer:

2. \( \log 14 = \log (2 \cdot 7) = \)

✏️ Your Answer:

3. \( \log \frac{7}{3} = \)

✏️ Your Answer:

4. \( \log \frac{3}{2} = \)

✏️ Your Answer:

Created by Pak Syah — SMAN 10 Samarinda

Properties (v) & (vi)

Property (v) — \( a^{{}^a \log b} = b \)

Example: \( 3^{{}^3 \log 7} = 7 \)

This is the definition of a logarithm in reverse!

Property (vi) — \( {}^{a^m} \log b^n = \frac{n}{m} \cdot {}^a \log b \)

Example 1: \( {}^\sqrt{2} \log 8 = {}^{2^{\frac{1}{2}}} \log 2^3 \)

Example 2: \( {}^5 \log 625 = {}^5 \log 5^4 \)

Simplify the following:

1. \( 4^{{}^4 \log 9} = \)

✏️ Your Answer:

2. \( {}^\sqrt{3} \log 81 = \)

✏️ Your Answer:

3. \( {}^2 \log 32 = \)

✏️ Your Answer:

4. \( {}^4 \log 64 = \)

✏️ Your Answer:

Created by Pak Syah — SMAN 10 Samarinda

Property (vii) — Change of Base

\( {}^a \log b = \frac{{}^p \log b}{{}^p \log a} \), with \( p > 0, p \neq 1 \)

Example

If \( {}^2 \log 3 = p \) and \( {}^2 \log 5 = q \), find \( {}^{12} \log 20 \) in terms of \( p \) and \( q \).

Express the following in terms of \( p \) and \( q \), given \( {}^2 \log 3 = p \) and \( {}^2 \log 5 = q \):

1. \( {}^2 \log 15 = \)

✏️ Your Answer:

2. \( {}^{6} \log 10 = \)

✏️ Your Answer:

3. \( {}^{12} \log 20 = \)

✏️ Your Answer:

Created by Pak Syah — SMAN 10 Samarinda

Consequences of Property (vii)

1) \( {}^a \log b = \frac{1}{{}^b \log a} \)

Example: \( {}^2 \log 8 = \frac{1}{{}^8 \log 2} \)

2) \( {}^a \log b \cdot {}^b \log c = {}^a \log c \)

Example: \( {}^2 \log 3 \cdot {}^3 \log 5 = {}^2 \log 5 \)

Evaluate the following:

1. \( {}^2 \log 3 \cdot {}^3 \log 4 = \)

✏️ Your Answer:

2. \( {}^5 \log 7 \cdot {}^7 \log 25 = \)

✏️ Your Answer:

3. \( {}^3 \log 5 \cdot {}^5 \log 9 = \)

✏️ Your Answer:

4. \( {}^4 \log 6 \cdot {}^6 \log 16 = \)

✏️ Your Answer:

Created by Pak Syah — SMAN 10 Samarinda

Real-World Application

Earthquake Magnitude (Richter Scale)

The intensity of an earthquake is measured using the Richter scale:

\( M = \log \frac{I}{I_0} \)

where \( I \) is the earthquake intensity and \( I_0 \) is a reference intensity.

Example

If earthquake A has \( I = 10^6 I_0 \) and earthquake B has \( I = 10^8 I_0 \), find the difference in their Richter scale magnitudes.

Solve the following:

1. If earthquake C has \( I = 10^{4.5} I_0 \), what is its Richter magnitude?

✏️ Your Answer:

2. If earthquake D is 100 times more intense than earthquake C, what is its magnitude?

✏️ Your Answer:

3. The difference between earthquake X and earthquake Y is 2 on the Richter scale. How many times more intense is earthquake Y?

✏️ Your Answer:

Created by Pak Syah — SMAN 10 Samarinda

CER — Claim, Evidence, Reasoning

Prompt

"Which property of logarithms is most useful for simplifying expressions?"

📌 Claim (Your statement):

✏️ Write your claim here:

📊 Evidence (Data/example to support):

✏️ Write your evidence here:

🧠 Reasoning (Explain the connection):

✏️ Write your reasoning here:

Created by Pak Syah — SMAN 10 Samarinda

Summary & Practice

Properties

(i) \( {}^a \log 1 = 0 \)
(ii) \( {}^a \log a = 1 \)
(iii) \( {}^a \log (bc) = {}^a \log b + {}^a \log c \)
(iv) \( {}^a \log \frac{b}{c} = {}^a \log b - {}^a \log c \)

Properties

(v) \( a^{{}^a \log b} = b \)
(vi) \( {}^{a^m} \log b^n = \frac{n}{m} \cdot {}^a \log b \)
(vii) \( {}^a \log b = \frac{{}^p \log b}{{}^p \log a} \)
Consequence: \( {}^a \log b \cdot {}^b \log c = {}^a \log c \)

Challenge: Simplify \( {}^2 \log 3 \cdot {}^3 \log 4 \cdot {}^4 \log 5 \cdot {}^5 \log 6 \)

✏️ Your Answer:

Created by Pak Syah — SMAN 10 Samarinda

Concept Integration

                    DRIVING QUESTION
                          │
                          ▼
           "How to solve logarithm problems?"
                          │
           ┌──────────────┼──────────────┐
           ▼              ▼              ▼
      PROPERTY (i)   PROPERTY (iii)   PROPERTY (vii)
      Log 1 = 0     Product → +      Change Base
      Log a = 1     Quotient → -     Inverse Property
                          │
                          ▼
                    SIMPLIFICATION
                          │
                          ▼
                   LOGARITHM VALUE
      

In your own words, explain how properties (iii) and (vii) work together:

✏️ Your Explanation:

Created by Pak Syah — SMAN 10 Samarinda

Reflection

🎓

"Mathematics is like a muscle — the more you use it, the stronger it gets."

Keys to mastering logarithms:

Memorize the 7 basic properties
Practice different types of problems
Look for patterns in each problem
Don't give up on challenging problems!

🌟 Reflection Question:

Which property do you find most useful? Which property is the most challenging? Why?

✏️ Write your reflection here:

Created by Pak Syah — Mathematics Teacher at SMAN 10 Samarinda

SMARIMATIKA