Worked problems for condensing logarithms using all properties

📚 Understanding Logarithmic Condensation

Condensing logarithms involves using the properties of logarithms to combine multiple logarithmic expressions into a single logarithm. This is the reverse process of expanding logarithms and is crucial in simplifying equations and solving for unknowns.

📜 History and Background

Logarithms were invented by John Napier in the early 17th century as a means to simplify calculations. They were quickly adopted by scientists and engineers for their ability to convert multiplication and division into addition and subtraction. The properties used for condensing logarithms are fundamental to their application.

🔑 Key Principles and Properties

These are the essential properties you'll need:

• ➕ Product Rule: log_b(x) + log_b(y) = log_b(xy)
• ➖ Quotient Rule: log_b(x) - log_b(y) = log_b(x/y)
• 🔢 Power Rule: n * log_b(x) = log_b(xⁿ)

💡 Step-by-Step Guide to Condensing Logarithms

1. 1️⃣ Apply the Power Rule: Move any coefficients in front of the logarithms as exponents of the arguments inside the logarithm.
2. 2️⃣ Apply the Product Rule: Combine logarithms connected by addition into a single logarithm by multiplying their arguments.
3. 3️⃣ Apply the Quotient Rule: Combine logarithms connected by subtraction into a single logarithm by dividing their arguments.
4. 4️⃣ Simplify: If possible, simplify the resulting expression.

✍️ Worked Examples

Example 1

Condense: $2log(x) + 3log(y) - log(z)$

• 🍎 Power Rule: $log(x^2) + log(y^3) - log(z)$
• ➕ Product Rule: $log(x^2y^3) - log(z)$
• ➗ Quotient Rule: $log(\frac{x^2y^3}{z})$

Example 2

Condense: $log_2(8) + log_2(5) - log_2(4)$

• ➕ Product Rule: $log_2(8 * 5) - log_2(4) = log_2(40) - log_2(4)$
• ➗ Quotient Rule: $log_2(\frac{40}{4}) = log_2(10)$

Example 3

Condense: $3log_5(x) - \frac{1}{2}log_5(y) + 4log_5(z)$

• 🍎 Power Rule: $log_5(x^3) - log_5(y^{\frac{1}{2}}) + log_5(z^4)$ which equals $log_5(x^3) - log_5(\sqrt{y}) + log_5(z^4)$
• ➕ Product Rule: $log_5(x^3z^4) - log_5(\sqrt{y})$
• ➗ Quotient Rule: $log_5(\frac{x^3z^4}{\sqrt{y}})$

Example 4

Condense: $log(12) - log(3) + 2log(x)$

• 🍎Power Rule: $log(12) - log(3) + log(x^2)$
• ➕ Product Rule: $log(12) + log(x^2) - log(3) = log(12x^2) - log(3)$
• ➗ Quotient Rule: $log(\frac{12x^2}{3}) = log(4x^2)$

Example 5

Condense: $\frac{1}{3}log_b(x) + 5log_b(y) - log_b(z)$

• 🍎Power Rule: $log_b(x^{\frac{1}{3}}) + log_b(y^5) - log_b(z)$ which equals $log_b(\sqrt[3]{x}) + log_b(y^5) - log_b(z)$
• ➕ Product Rule: $log_b(\sqrt[3]{x} * y^5) - log_b(z)$
• ➗ Quotient Rule: $log_b(\frac{y^5\sqrt[3]{x}}{z})$

Example 6

Condense: $4log(x) - log(y) - 6log(z)$

• 🍎Power Rule: $log(x^4) - log(y) - log(z^6)$
• ➖ Quotient Rule (applied twice): $log(\frac{x^4}{y}) - log(z^6) = log(\frac{x^4}{yz^6})$

Example 7

Condense: $2log_3(x + 1) + log_3(x) - log_3(5)$

• 🍎Power Rule: $log_3((x + 1)^2) + log_3(x) - log_3(5)$
• ➕ Product Rule: $log_3((x + 1)^2 * x) - log_3(5)$
• ➗ Quotient Rule: $log_3(\frac{x(x + 1)^2}{5})$

📝 Practice Quiz

Condense the following expressions:

1. ❓ $log(5) + log(x) - log(y)$
2. ❓ $2log(a) - 3log(b) + log(c)$
3. ❓ $\frac{1}{2}log(x) + log(y) - 2log(z)$

Answers:

1. ✅ $log(\frac{5x}{y})$
2. ✅ $log(\frac{a^2c}{b^3})$
3. ✅ $log(\frac{\sqrt{x}y}{z^2})$

🎯 Conclusion

Mastering the condensation of logarithms requires a solid understanding of their properties. By practicing these steps and examples, you'll become proficient in simplifying complex logarithmic expressions. Keep practicing, and you'll conquer those logs! 💪