A science communicator is explaining the exponential growth of bacteria in a petri dish. If the growth is modeled by the equation \( 5e^{kt} = 100 \), where \( k \) and \( t \) are positive, solve for \( t \) when \( k = 0.3 \).

["Title: How Exponential Bacterial Growth Works: Solving for Time with the Equation ( 5e^{0.3t} = 100 )", "Understanding bacterial growth is a fascinating topic in biology and science communication—and today, we’re diving into a classic example: how exponential growth models bacterial populations in a petri dish. Whether you’re a student, educator, or science enthusiast, seeing how equations like ( 5e^{kt} = 100 ) explain real-world phenomena helps unlock the beauty of science.", "### What Is Exponential Growth in Bacteria?", "Bacterial populations often grow exponentially under ideal conditions—plenty of nutrients, no predators, and optimal temperature. This growth isn’t linear; instead, the number of bacteria doubles at consistent time intervals, described mathematically by exponential functions.", "In one common model, the population ( P(t) ) at time ( t ) is given by:\n[\nP(t) = P_0 e^{kt}\n]\nwhere ( P_0 ) is the initial population, ( k ) is the growth rate constant, and ( t ) is time.", "In our example, the equation is simplified and given as:\n[\n5e^{kt} = 100\n]\nHere, 5 represents the initial bacterial count when ( t = 0 ), and ( k = 0.3 ) is the growth constant. Our goal is to solve for ( t ) when the population reaches 100 bacteria.", "### Step-by-Step Solution", "Start with the equation:\n[\n5e^{0.3t} = 100\n]", "Step 1: Divide both sides by 5 to isolate the exponential term.\n[\ne^{0.3t} = \frac{100}{5} = 20\n]", "Step 2: Take the natural logarithm (ln) of both sides to eliminate the exponential.\n[\n\ln(e^{0.3t}) = \ln(20)\n]\nUsing the property ( \ln(e^x) = x ), this simplifies to:\n[\n0.3t = \ln(20)\n]", "Step 3: Solve for ( t ).\n[\nt = \frac{\ln(20)}{0.3}\n]", "Step 4: Calculate using approximate values:\n( \ln(20) \approx 2.9957 )\n[\nt \approx \frac{2.9957}{0.3} \approx 9.985 \ ext{ hours}\n]", "### Final Result", "So, when ( k = 0.3 ), the time ( t ) required for the bacterial population to grow from 5 to 100 bacteria is approximately 10 hours.", "This kind of exponential growth explains why bacterial cultures must be monitored closely over time—small changes in conditions or initial populations can rapidly lead to exponential increases.", "Science communicators play a vital role in translating such equations into intuitive understanding, turning abstract math into meaningful insights about life at the microscopic level. Whether you’re culturing for research or inspiring curiosity, knowing how to solve equations like ( 5e^{kt} = 100 ) deepens your connection to biology and the power of mathematical modeling.", "---\nKeywords: bacterial growth, exponential growth, ( 5e^{kt} = 100 ), science communication, exponential equation, ( k = 0.3 ), ( t ) solution, petri dish biology"]