Mathematics 9709/33 — May/June 2019

Walkthrough

We are asked to solve the exponential equation

using logarithms.

1. Take logarithms of both sides. Any base works, but the natural logarithm $\ln$ is the most common and is acceptable here.
2. Apply the power law of logarithms on the left and the product law on the right. This transforms the exponential equation into a linear equation in $x$ because the unknown index can now be brought down in front of the logarithm.
3. Expand the left-hand side and collect terms involving $x$ on one side, with constant terms on the other.
4. Factor out $x$ and divide by its coefficient to obtain the exact expression for $x$.
5. Use a calculator to evaluate the expression to 3 decimal places, giving $0.666$.

Key Takeaways

• The main skill is applying the laws of logarithms to bring an unknown index down from an exponent.
• The product law allows a product inside a logarithm to be split, and the power law allows an exponent to be moved in front of the logarithm.
• Solving an exponential equation by taking logarithms turns it into a linear equation.
• When the unknown appears in more than one exponent, both occurrences must be handled carefully and collected on the same side.

Common Mistakes

• Forgetting to take logarithms of the entire right-hand side, e.g. writing $\ln\left(4(7^x)\right)$ as $\ln 4 + \ln(7^x)$ is correct, but writing it as $\ln 4 \cdot \ln(7^x)$ is wrong.
• Using the power law incorrectly: for example, forgetting the bracket on the left and writing $3-2x\ln 5$ instead of $(3-2x)\ln 5$.
• Sign errors when rearranging the linear terms, especially when moving $2x\ln 5$ across the equals sign.
• Rounding too early, before the final calculation, which may give a slightly different 3-decimal answer.

Things to Be Careful About

• The mark scheme requires a correct linear equation such as

before solving the final answer. Unsupported final answers may not receive full marks.

• Keep the expression exact until the very end, then evaluate. Do not round intermediate values.
• Make sure the final answer is given to exactly 3 decimal places: $0.666$.