A scientist is conducting an experiment involving bacteria. The equation N = 100 3^...

1. Set up the equation: We are given the formula for the number of bacteria $N$ after $t$ days as $N = 100 \cdot 3^t$. We want to find the number of days $t$ it takes for the number of bacteria to reach 2,700. So, we set $N = 2700$.
$$2700 = 100 \cdot 3^t$$
2. Isolate the exponential term: To solve for $t$, we first need to isolate the term $3^t$. We can do this by dividing both sides of the equation by 100.
$$\frac{2700}{100} = \frac{100 \cdot 3^t}{100}$$
$$27 = 3^t$$
3. Solve for t using logarithms: Now we have a simple exponential equation. To solve for $t$, we can take the logarithm of both sides. It is convenient to use the logarithm with base 3, since the base of the exponential term is 3.
$$\log_3(27) = \log_3(3^t)$$
4. Simplify and find t: Using the property of logarithms that $\log_b(b^x) = x$, the right side simplifies to $t$. For the left side, we need to find what power we must raise 3 to in order to get 27. We know that $3^3 = 27$.
$$3 = t$$