Expanding Logarithmic Expressions: Sums and Differences

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

• a) $\log_{25}(25xy)$

• b) $\log_{4}\left(\frac{7}{8}\right)$

• c) $\log_{5}\left(\frac{4}{3}\right)$

• d) $\log_{12}(100n^2)$

• e) $\ln(x^2 \cdot \sqrt{x+3})$

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: $\log_b(MN) = \log_b(M) + \log_b(N)$

• Quotient Rule: $\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$

• Power Rule: $\log_b(M^k) = k \cdot \log_b(M)$

• Natural Logarithm: $\ln(x)$ is the logarithm base $e$.

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: $\log_b(MN) = \log_b(M) + \log_b(N)$.

3. Apply the quotient rule to split quotients inside the logarithm into differences: $\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$.

4. Apply the power rule to bring exponents in the argument out in front: $\log_b(M^k) = k \cdot \log_b(M)$.

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!