Solved Examples: Expanding Logarithmic Expressions with Properties

📚 Quick Study Guide

• 🔑 Product Rule: The logarithm of a product is the sum of the logarithms. That is, $\log_b(MN) = \log_b(M) + \log_b(N)$.
• ➗ Quotient Rule: The logarithm of a quotient is the difference of the logarithms. That is, $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$.
• ⚡ Power Rule: The logarithm of a power is the exponent times the logarithm. That is, $\log_b(M^k) = k \log_b(M)$.
• ➕ Combining Rules: You may need to use multiple rules in a single problem. Remember to apply the power rule before the product or quotient rules where applicable.
• 💡 Important Note: These rules only apply when the logarithms have the same base.

Practice Quiz

1. What is the expansion of $\log_2(8x^5)$?
   1. $\log_2(8) + 5\log_2(x)$
   2. $3 + 5\log_2(x)$
   3. $\log_2(8) + \log_2(x^5)$
   4. All of the above
2. Expand the expression: $\log(\frac{a^2}{b})$
   1. $2\log(a) - \log(b)$
   2. $\frac{2\log(a)}{\log(b)}$
   3. $2\log(a) + \log(b)$
   4. $\log(a^2) + \log(b)$
3. Expand: $\ln(\sqrt[3]{x})$
   1. $3\ln(x)$
   2. $\frac{1}{3}\ln(x)$
   3. $\ln(3x)$
   4. $\ln(x^3)$
4. Which expression is equivalent to $\log_5(25x)$?
   1. $2 + \log_5(x)$
   2. $\log_5(25) \cdot \log_5(x)$
   3. $2 \log_5(x)$
   4. $\log_5(25) - \log_5(x)$
5. Expand: $\log_b(\frac{x^3y}{z^2})$
   1. $3\log_b(x) + \log_b(y) - 2\log_b(z)$
   2. $3\log_b(x) + \log_b(y) + 2\log_b(z)$
   3. $\log_b(x^3) + \log_b(y) + \log_b(z^2)$
   4. $\log_b(x^3yz^{-2})$
6. What is the expanded form of $\log_4(\frac{16}{\sqrt{x}})$?
   1. $2 - \frac{1}{2}\log_4(x)$
   2. $4 - \frac{1}{2}\log_4(x)$
   3. $2 + \frac{1}{2}\log_4(x)$
   4. $4 + \frac{1}{2}\log_4(x)$
7. Expand $\log_3(9x^2y^3)$
   1. $2 + 2\log_3(x) + 3\log_3(y)$
   2. $2 + 2\log_3(x) + \log_3(y^3)$
   3. $\log_3(9) + \log_3(x^2) + \log_3(y^3)$
   4. All of the above