Logarithmic Expansion Examples with Detailed Solutions

📚 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: $\log_b(M^k) = k \cdot \log_b(M)$
• 💡 Change of Base Rule: $\log_b(M) = \frac{\log_c(M)}{\log_c(b)}$
• ➕ Logarithms expand when a single logarithm contains multiple terms through multiplication, division, or exponents.

📝 Practice Quiz

1. What is the expansion of $\log_2(8x)$?
   1. $\log_2(8) + \log_2(x)$
   2. $3\log_2(x)$
   3. $\log_2(8) - \log_2(x)$
   4. $\frac{\log_2(8)}{\log_2(x)}$
2. Expand the following logarithm: $\log(100a^2b)$
   1. $2 + 2\log(a) + \log(b)$
   2. $\log(100) + \log(a) + \log(b)$
   3. $2 + \log(a^2) + \log(b)$
   4. $2 + 2\log(10a) + \log(b)$
3. Which expression is equivalent to $\ln(\frac{x^3}{y^2})$?
   1. $3\ln(x) - 2\ln(y)$
   2. $\frac{3}{2} \ln(\frac{x}{y})$
   3. $3\ln(x) + 2\ln(y)$
   4. $\ln(3x) - \ln(2y)$
4. Expand: $\log_5(\sqrt[3]{25x})$
   1. $\frac{2}{3} + \frac{1}{3}\log_5(x)$
   2. $\frac{1}{3}\log_5(25) + \log_5(x)$
   3. $\frac{2}{3} \log_5(x)$
   4. $\frac{1}{3}(\log_5(25) + \log_5(x))$
5. What is the expanded form of $\log_b(\frac{x^4}{\sqrt{y}})$?
   1. $4\log_b(x) - \frac{1}{2}\log_b(y)$
   2. $\frac{4\log_b(x)}{\frac{1}{2}\log_b(y)}$
   3. $4\log_b(x) - 2\log_b(y)$
   4. $4\log_b(x) + \frac{1}{2}\log_b(y)$
6. Simplify: $\log_3(9a^5b^{-2})$
   1. $2 + 5\log_3(a) - 2\log_3(b)$
   2. $\log_3(9) + 5\log_3(a) + 2\log_3(b)$
   3. $2 + 5\log_3(a) + 2\log_3(b)$
   4. $9 + 5\log_3(a) - 2\log_3(b)$
7. Expand the logarithm: $\ln(\frac{e^2\sqrt{x}}{y^3})$
   1. $2 + \frac{1}{2}\ln(x) - 3\ln(y)$
   2. $2 + 2\ln(x) - 3\ln(y)$
   3. $e^2 + \frac{1}{2}\ln(x) - 3\ln(y)$
   4. $2 + \sqrt{x} - 3\ln(y)$