IB Mathematics SL 1.7 Laws of exponents with rational exponents AA SL Paper 1- Exam Style Questions- New Syllabus
Question
Write each of the following can be expressed in the form \( \ln k \), where \( k \in \mathbb{Z}^+ \).
(a) \( \ln 3 + \ln 4 \)
(b) \( 3 \ln 2 \)
(c) \( -\ln \frac{1}{2} \)
Most-appropriate topic codes (IB Mathematics Analysis and Approaches SL 2025):
• SL 1.7: Laws of logarithms — parts (a), (b), (c)
• SL 2.9: Logarithmic functions as inverses of exponentials — conceptual link
• Prior learning: Manipulation of algebraic expressions — supporting skill
• SL 2.9: Logarithmic functions as inverses of exponentials — conceptual link
• Prior learning: Manipulation of algebraic expressions — supporting skill
▶️ Answer/Explanation
(a)
Using the product law of logarithms: \( \ln a + \ln b = \ln(ab) \).
\( \ln 3 + \ln 4 = \ln(3 \times 4) = \ln 12 \).
\( \boxed{\ln 12} \)
(b)
Using the power law of logarithms: \( p \ln a = \ln(a^p) \).
\( 3 \ln 2 = \ln(2^3) = \ln 8 \).
\( \boxed{\ln 8} \)
(c)
Method 1: Using power law: \( -\ln \frac{1}{2} = \ln\left( \left( \frac{1}{2} \right)^{-1} \right) = \ln 2 \).
Method 2: Using quotient law: \( -\ln \frac{1}{2} = \ln 1 – \ln \frac{1}{2} = \ln\left( \frac{1}{1/2} \right) = \ln 2 \).
\( \boxed{\ln 2} \)