Mathematics 9709/32 — February/March 2016

Walkthrough

This problem tests your command of the laws of logarithms and the fact that the logarithm is a one-to-one function, which lets us "cancel" logarithms on both sides once the arguments are made equal.

Step 1: Apply the power law. The term $2 \ln x$ has a coefficient of 2. The power law $a \ln b = \ln(b^a)$ lets us rewrite it as $\ln(x^2)$. This is the first mark in the scheme (M1), and it is essential because it turns the whole right-hand side into a sum of two logarithms, which we can then merge.

Step 2: Apply the product law. With $2 \ln x$ replaced by $\ln(x^2)$, the right-hand side is $\ln(x^2) + \ln 4$. The product law $\ln a + \ln b = \ln(ab)$ combines these into the single logarithm $\ln(4x^2)$. Now both sides consist of one logarithm of a single argument.

Step 3: Remove the logarithms. Since $\ln$ is injective (one-to-one), $\ln A = \ln B$ implies $A = B$. Thus we can drop the logarithms and equate the arguments: $x^2 + 4 = 4x^2$. This is the A1 mark.

Step 4: Solve for $x$. Subtract $x^2$ from both sides to get $4 = 3x^2$, so $x^2 = \frac{4}{3}$ and $x = \pm \frac{2}{\sqrt{3}}$.

Step 5: Apply the domain restriction. The original equation contains $\ln x$, which is only defined when $x > 0$. Therefore the negative root $x = -\frac{2}{\sqrt{3}}$ must be discarded, leaving the unique solution $x = \frac{2}{\sqrt{3}}$. This is the final A1 mark.

Key Takeaways

• The three fundamental laws of logarithms (power, product, quotient) are tools for combining and simplifying logarithmic expressions.
• Because the logarithm is a one-to-one function, equal logarithms force equal arguments — this is the standard way to "remove" logarithms from an equation.
• Every logarithm in an equation imposes a domain restriction: its argument must be positive. Always check your final answers against these restrictions.

Common Mistakes

• Skipping the combination of logarithms. Some students remove logarithms immediately, incorrectly writing $x^2 + 4 = 2x + 4$ or similar. You must first merge the two logarithms on the right into one.
• Sign error in collecting terms. When moving $x^2$ across, the sign can flip incorrectly. Be careful: $x^2 + 4 = 4x^2$ leads to $4 = 3x^2$.
• Taking the negative root. The mark scheme expects the single positive answer; the negative root is invalid because $\ln x$ is undefined for $x \leq 0$.

Things to Be Careful About

• The domain of $\ln x$ requires $x > 0$ — this is why $x = -\frac{2}{\sqrt{3}}$ is rejected.
• The power law must be applied first; otherwise the sum on the right cannot be combined into a single logarithm.
• When giving the answer in "exact form", do not rationalise the denominator unnecessarily — $\frac{2}{\sqrt{3}}$ is the expected form, though $\frac{2\sqrt{3}}{3}$ is an acceptable equivalent.