Q3. Expand each of the following expressions as much as possible into a sum or difference of logarithms.

• a)

• b)

• c)

• d)

• e)

Background

Topic: Properties of Logarithms

This question is testing your ability to use logarithmic properties to rewrite expressions as sums or differences. This is a foundational skill in calculus and business mathematics, especially for simplifying expressions before differentiation or integration.

Key Terms and Formulas

• Product Rule:

• Quotient Rule:

• Power Rule:

• Natural Logarithm: is the logarithm base .

Step-by-Step Guidance

1. For each expression, identify whether it involves a product, quotient, or power inside the logarithm.

2. Apply the product rule to split products inside the logarithm into sums: .

3. Apply the quotient rule to split quotients inside the logarithm into differences: .

4. Apply the power rule to bring exponents in the argument out in front: .

5. For expressions involving roots, rewrite the root as a fractional exponent before applying the power rule.

Try solving on your own before revealing the answer!