Solved examples: Condensing logarithmic expressions with multiple terms

📚 Quick Study Guide

• 🔑 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: ⚡ $\log_b(M^k) = k \cdot \log_b(M)$
• 💡 Combining Terms: Use these rules in reverse to condense. For example, $2\log(x) + 3\log(y) - \log(z) = \log(\frac{x^2y^3}{z})$
• 🧐 Common Logarithm: If the base isn't written, it's base 10.

Practice Quiz

1. What is the condensed form of $\log(5) + \log(x) - \log(2)$?
   1. $\log(5x/2)$
   2. $\log(5 + x - 2)$
   3. $\log(5x - 2)$
   4. $\log(5/2x)$
2. Condense the expression: $2\log_b(x) + 3\log_b(y)$
   1. $\log_b(x^2y^3)$
   2. $\log_b(x^2 + y^3)$
   3. $\log_b(6xy)$
   4. $\log_b((x+y)^5)$
3. What is the condensed form of $3\log(x) - \frac{1}{2}\log(y) + \log(z)$?
   1. $\log(\frac{x^3z}{\sqrt{y}})$
   2. $\log(\frac{x^3}{z\sqrt{y}})$
   3. $\log(\frac{3x}{2\sqrt{y}z})$
   4. $\log(x^3 - \sqrt{y} + z)$
4. Condense: $\log(a) - \log(b) + \log(c) - \log(d)$
   1. $\log(\frac{ac}{bd})$
   2. $\log(\frac{a-b}{c-d})$
   3. $\log(a-b+c-d)$
   4. $\log(\frac{ad}{bc})$
5. Simplify: $\frac{1}{2} \log(9) + 2 \log(2) - \log(3)$
   1. $\log(\frac{4 \cdot 3}{3}) = \log(4)$
   2. $\log(\frac{\sqrt{9} + 4}{3})$
   3. $\log(\frac{9}{4 \cdot 3})$
   4. $\log(\frac{\sqrt{9} \cdot 4}{3})$
6. Condense the expression: $\log_2(8) + \log_2(4) - \log_2(2)$
   1. $\log_2(16)$
   2. $\log_2(10)$
   3. $\log_2(32)$
   4. $\log_2(4)$
7. What is the condensed form of $\log(100) - \log(10) + \log(1)$?
   1. $\log(10)$
   2. $\log(91)$
   3. $\log(11)$
   4. $\log(1000)$