Mathematics 9709/21 — May/June 2017

Walkthrough

The equation $5^x = 3^{4y}$ has the unknown in the exponents. To bring the exponents down, take logarithms of both sides. Any base is valid, but natural logarithms are standard. This gives $\ln(5^x) = \ln(3^{4y})$.

Next, use the power law of logarithms, $\ln(a^b) = b\ln a$, to rewrite each side as $x\ln 5$ and $4y\ln 3$. Since these two expressions are equal, we have $x\ln 5 = 4y\ln 3$.

Now solve for $y$ by dividing both sides by $4\ln 3$:

This is of the form $y = mx$, so $m = \frac{\ln 5}{4\ln 3}$. Evaluating this on a calculator gives approximately $0.366$ to 3 significant figures.

Key Takeaways

• Logarithms are used to bring exponents down so that equations with unknown indices can be solved.
• The power law $\ln(a^b) = b\ln a$ must be applied to both sides of the equation.
• An equation of the form $a^x = b^y$ can often be rewritten as a linear relationship $y = mx$ using logarithms.

Common Mistakes

• Forgetting to apply the power law to both sides of the equation.
• Dividing by the wrong factor, for example writing $y = \frac{4\ln 3}{\ln 5}x$ instead of $y = \frac{\ln 5}{4\ln 3}x$.
• Mixing logarithm bases in the same equation, such as using $\ln$ on one side and $\log_{10}$ on the other.
• Giving the exact expression but not rounding to 3 significant figures as requested.

Things to Be Careful About

• Keep the same base of logarithm on both sides throughout.
• The factor $4$ multiplies $\ln 3$, so it stays in the denominator: $m = \frac{\ln 5}{4\ln 3}$, not $\frac{4\ln 3}{\ln 5}$.
• The numerical value is $\frac{\ln 5}{4\ln 3} \approx \frac{1.6094}{4.3944} \approx 0.3662$, so to 3 significant figures the answer is $0.366$.
• The question asks for the value of the constant $m$ correct to 3 significant figures, so the final answer should be the rounded decimal $0.366$.