Step-by-Step Solution
Key Concept: Divide the equation by 9^t to create a single-variable equation in terms of (2/3)^t, then recognize that the resulting expression equals the golden ratio conjugate, which is solvable via logarithms.
<p><strong>Step 1:</strong> Divide both sides by 9^t to get a normalized equation:</p><p>$$\frac{4^t}{9^t} + \frac{6^t}{9^t} = 1$$</p><p><strong>Step 2:</strong> Rewrite using exponent rules:</p><p>$$\left(\frac{4}{9}\right)^t + \left(\frac{6}{9}\right)^t = 1$$</p><p>$$\left(\frac{2}{3}\right)^{2t} + \left(\frac{2}{3}\right)^t = 1$$</p><p><strong>Step 3:</strong> Let u = (2/3)^t to obtain a quadratic:</p><p>$$u^2 + u - 1 = 0$$</p><p><strong>Step 4:</strong> Apply the quadratic formula:</p><p>$$u = \frac{-1 \pm \sqrt{1+4}}{2} = \frac{-1 \pm \sqrt{5}}{2}$$</p><p><strong>Step 5:</strong> Since u = (2/3)^t > 0, we take the positive root:</p><p>$$u = \frac{-1 + \sqrt{5}}{2} = \frac{\sqrt{5}-1}{2}$$</p><p><strong>Step 6:</strong> Solve for t using logarithms:</p><p>$$(2/3)^t = \frac{\sqrt{5}-1}{2}$$</p><p>$$t \ln(2/3) = \ln\left(\frac{\sqrt{5}-1}{2}\right)$$</p><p>$$t = \frac{\ln\left(\frac{\sqrt{5}-1}{2}\right)}{\ln(2/3)} = \log_{2/3}\left(\frac{\sqrt{5}-1}{2}\right)$$</p><p>∴ Answer: <strong>t = \log_{2/3}\left(\dfrac{\sqrt{5}-1}{2}\right)</strong></p>
Correct Answer: t = \log_{2/3}\left(\dfrac{\sqrt{5}-1}{2}\right)