Mathematics 9709/21 — May/June 2019

Walkthrough

Start by recognising that the expression has the form $\ln A - \ln B$. The logarithm law $\ln A - \ln B = \ln\left(\frac{A}{B}\right)$ lets us combine the two logarithms into a single logarithm of a quotient. This is the key step because it turns the difference of two logarithms into one expression that can then be simplified algebraically.

Next, factorise the numerator $x^3 - 4x$. Factor out the common factor $x$ to get $x(x^2 - 4)$, and then use the difference of two squares to write $x^2 - 4 = (x - 2)(x + 2)$. So $x^3 - 4x = x(x - 2)(x + 2)$.

The denominator $x^2 - 2x$ also has a common factor $x$, giving $x(x - 2)$.

Now substitute these factorised forms into the quotient:

The common factor $x(x - 2)$ cancels, provided $x(x - 2) \neq 0$, leaving $x + 2$.

Therefore the original difference of logarithms simplifies to $\ln(x + 2)$, which is the required identity.

Key Takeaways

• The logarithm subtraction law $\ln A - \ln B = \ln\left(\frac{A}{B}\right)$ is essential for combining logarithmic terms.
• Algebraic factorisation is often needed before a logarithmic expression can be simplified fully.
• Cancelling common factors is valid only when the factor is not zero; the identity holds on the common domain where the original logarithms are defined.

Common Mistakes

• Applying the subtraction law incorrectly, such as writing $\ln A - \ln B = \frac{\ln A}{\ln B}$ instead of $\ln\left(\frac{A}{B}\right)$.
• Failing to factorise the numerator fully, so the common factor is not seen.
• Cancelling terms that are not common factors, such as cancelling $x$ without also cancelling the corresponding factor in the denominator.
• Forgetting to state that the cancellation requires $x(x - 2) \neq 0$.

Things to Be Careful About

• The expression is an identity only where both original logarithms are defined, so the arguments $x^3 - 4x$ and $x^2 - 2x$ must be positive and $x(x - 2) \neq 0$.
• The mark scheme awards M1 for using the logarithm subtraction law, B1 for factorising or algebraic division, and A1 for obtaining the final answer $\ln(x + 2)$ with no errors seen.
• Be careful with the difference of two squares: $x^2 - 4 = (x - 2)(x + 2)$, not $(x - 2)^2$.