1. **State the problem:** We need to find the value of $t$, a positive real number, given the equation: $$\log_3 t + \log_9 t + \log_{27} t + \log_{81} t = 10$$ We want the answer in the form $3^r$ where $r$ is a rational number. 2. **Recall the formula:** $$\log_a b = \frac{\log_c b}{\log_c a}$$ This means we can express all logarithms with the same base for easier calculation. 3. **Express all logs with base 3:** Since $9 = 3^2$, $27 = 3^3$, and $81 = 3^4$, we have: $$\log_9 t = \frac{\log_3 t}{\log_3 9} = \frac{\log_3 t}{2}$$ $$\log_{27} t = \frac{\log_3 t}{\log_3 27} = \frac{\log_3 t}{3}$$ $$\log_{81} t = \frac{\log_3 t}{\log_3 81} = \frac{\log_3 t}{4}$$ 4. **Substitute into the original equation:** $$\log_3 t + \frac{\log_3 t}{2} + \frac{\log_3 t}{3} + \frac{\log_3 t}{4} = 10$$ Let $x = \log_3 t$ for simplicity. 5. **Combine terms:** $$x + \frac{x}{2} + \frac{x}{3} + \frac{x}{4} = 10$$ Find common denominator 12: $$\frac{12x}{12} + \frac{6x}{12} + \frac{4x}{12} + \frac{3x}{12} = 10$$ $$\frac{12x + 6x + 4x + 3x}{12} = 10$$ $$\frac{25x}{12} = 10$$ 6. **Solve for $x$:** Multiply both sides by 12: $$\cancel{\frac{25x}{\cancel{12}}} \times 12 = 10 \times 12$$ $$25x = 120$$ Divide both sides by 25: $$x = \frac{120}{25}$$ Simplify fraction: $$x = \frac{24}{5}$$ 7. **Recall $x = \log_3 t$, so:** $$\log_3 t = \frac{24}{5}$$ Rewrite in exponential form: $$t = 3^{\frac{24}{5}}$$ **Final answer:** $$t = 3^{\frac{24}{5}}$$