Solve for t: 3^(t/5) = 9. What is the time in years?

The population growth is modeled by the equation:

$$3^{t/5} = 9$$

We know that 9 can be written as $3^2$.

$$3^{t/5} = 3^2$$

Since the bases are the same, the exponents must be equal:

$$\frac{t}{5} = 2$$

Multiply both sides by 5:

$$t = 2 \times 5$$ $$t = 10$$

Alternatively, using logarithms as requested:

Using the common logarithm (base 10):

$$\log(3^{t/5}) = \log(9)$$

Since $9 = 3^2$, we can write $\log(9)$ as $2 \log(3)$:

$$t = \frac{5 \cdot (2 \log(3))}{\log(3)}$$ $$t = 5 \times 2$$ $$t = 10$$

It takes 10 years for the population to reach 9 times its original size.

The time taken is $\text{10 years}$.