A-Level Mathematics May/June 2024 Q1: Solve the equation 8^(3-6x) = 4×5^(-2x). Give your answer correct to 3 decimal places.

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  The given equation is $8^{3-6x} = 4 \times 5^{-2x}$.

  To solve for $x$, we take the natural logarithm (ln) of both sides of the equation.

  $\ln(8^{3-6x}) = \ln(4 \times 5^{-2x})$

  Using the logarithm law $\ln(ab) = \ln(a) + \ln(b)$ on the right-hand side:

  $\ln(8^{3-6x}) = \ln(4) + \ln(5^{-2x})$

  Now, using the logarithm law $\ln(a^p) = p\ln(a)$ on both sides:

  $(3-6x)\ln(8) = \ln(4) - 2x\ln(5)$

  The next step is to expand the bracket and rearrange the equation to make $x$ the subject.

  $3\ln(8) - 6x\ln(8) = \ln(4) - 2x\ln(5)$

  Group all terms containing $x$ on one side and the constant terms on the other side.

  $3\ln(8) - \ln(4) = 6x\ln(8) - 2x\ln(5)$

  Factor out $x$ from the terms on the right-hand side.

  $3\ln(8) - \ln(4) = x(6\ln(8) - 2\ln(5))$

  Finally, solve for $x$ by dividing.

  $x = \frac{3\ln(8) - \ln(4)}{6\ln(8) - 2\ln(5)}$

  Now, we calculate the numerical value using a calculator.

  $x = \frac{4.852030264...}{9.257760528...}$ $x = 0.524100...$

  Rounding the answer to 3 decimal places as required:

  $x = 0.524$

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  How the marks are awarded

  • B1 — Correctly taking logarithms of both sides and applying the product rule for logarithms to the right-hand side, resulting in the line: $\ln(8^{3-6x}) = \ln(4) + \ln(5^{-2x})$.
  • DB1 — Correctly applying the power rule for logarithms to at least one side of the equation, for example to get $(3-6x)\ln(8)$ or $-2x\ln(5)$.
  • B1 — Obtaining a correct linear equation in $x$ by correctly expanding brackets and rearranging terms, such as $x(6\ln(8) - 2\ln(5)) = 3\ln(8) - \ln(4)$.
  • B1 — Obtaining the final correct answer of $x = 0.524$, rounded to 3 decimal places, from a correct method.

  Common mistakes

  • Incorrectly applying the logarithm product rule, for example writing $\ln(4 \times 5^{-2x})$ as $\ln(4) \times \ln(5^{-2x})$.
  • Making sign errors when rearranging the equation to isolate $x$. For example, incorrectly moving $-6x\ln(8)$ to the right side as $-6x\ln(8)$ instead of changing the sign.
  • Incorrectly expanding brackets involving logarithms, for example writing $(3-6x)\ln(8)$ as $3\ln(8) - 6x$ instead of the correct $3\ln(8) - 6x\ln(8)$.
  • Rounding intermediate decimal values for logarithms too early, which can lead to an inaccurate final answer and loss of the final accuracy mark.

  Examiner tip: When solving an exponential equation where the bases cannot be easily unified, the standard technique is to take logarithms of both sides to convert the exponential terms into linear terms that you can then rearrange and solve.

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