Expanding vs. Condensing Logarithms: Understanding the key difference.

📚 Expanding vs. Condensing Logarithms: Understanding the Key Difference

Logarithms are a fundamental concept in mathematics, used to simplify complex calculations and solve exponential equations. The operations of expanding and condensing logarithms are crucial for manipulating logarithmic expressions. Let's explore the key differences between them.

Expanding Logarithms:

Expanding logarithms involves breaking down a single logarithmic expression into a sum or difference of multiple logarithmic terms. This process utilizes the properties of logarithms to separate the factors within the argument.

Condensing Logarithms:

Condensing logarithms is the reverse process of expanding. It involves combining multiple logarithmic terms into a single logarithmic expression. This simplification also relies on the properties of logarithms.

📝 Comparison Table: Expanding vs. Condensing Logarithms

🔑 Key Takeaways

• ➕ Expanding logarithms uses properties to split a single logarithm into multiple logarithms.
• ➖ Condensing logarithms uses properties to combine multiple logarithms into a single logarithm.
• 💡 Understanding these processes allows for effective manipulation and simplification of logarithmic expressions.

🧮 Properties of Logarithms

Both expanding and condensing rely on these logarithmic properties:

• ✖️ Product Rule: $\log_b(MN) = \log_b(M) + \log_b(N)$
• ➗ Quotient Rule: $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$
• ⚡ Power Rule: $\log_b(M^p) = p \cdot \log_b(M)$

✏️ Example Problems

Let's walk through a few examples to illustrate the process of expanding and condensing logarithms.

Expanding Example:

Expand the following logarithm: $\log( \frac{x^2y}{z})$

1. ✏️ Apply the quotient rule: $\log(x^2y) - \log(z)$
2. ✖️ Apply the product rule: $\log(x^2) + \log(y) - \log(z)$
3. ⚡ Apply the power rule: $2\log(x) + \log(y) - \log(z)$

Therefore, $\log( \frac{x^2y}{z}) = 2\log(x) + \log(y) - \log(z)$

Condensing Example:

Condense the following expression: $2\log(x) + 3\log(y) - \log(z)$

1. ⚡ Apply the power rule: $\log(x^2) + \log(y^3) - \log(z)$
2. ✖️ Apply the product rule: $\log(x^2y^3) - \log(z)$
3. ➗ Apply the quotient rule: $\log( \frac{x^2y^3}{z})$

Therefore, $2\log(x) + 3\log(y) - \log(z) = \log( \frac{x^2y^3}{z})$

✍️ Practice Quiz

Test your knowledge with these problems. Answers are provided below.

1. ❓ Expand: $\log(5x^3)$
2. ❓ Expand: $\log(\frac{a}{b^2})$
3. ❓ Expand: $\log(\sqrt{xy})$
4. ❓ Condense: $\log(2) + \log(x) + \log(y)$
5. ❓ Condense: $2\log(a) - 3\log(b)$
6. ❓ Condense: $\frac{1}{2}\log(x) + 3\log(y)$

✅ Answer Key

1. ✅ $\log(5) + 3\log(x)$
2. ✅ $\log(a) - 2\log(b)$
3. ✅ $\frac{1}{2}(\log(x) + \log(y))$
4. ✅ $\log(2xy)$
5. ✅ $\log(\frac{a^2}{b^3})$
6. ✅ $\log(y^3\sqrt{x})$