a schools gardening club is conducting a campaign to plant herbal plants in the community. the number of…

a schools gardening club is conducting a campaign to plant herbal plants in the community. the number of herbal plants grows exponentially over time and can be modeled by the function h(x)=50·e^x, where x is the number of weeks. write and evaluate a logarithmic equation to show how many weeks it takes to plant 500 herbal plants. round your answer to the nearest whole number of weeks. use the keypad to enter the answers in the boxes. more symbols can be found using the drop - down arrow at the top of the keypad. a logarithmic equation to find how many weeks it takes to plant 500 herbal plants is . it takes about weeks to plant 500 herbal plants.

a schools gardening club is conducting a campaign to plant herbal plants in the community. the number of herbal plants grows exponentially over time and can be modeled by the function h(x)=50·e^x, where x is the number of weeks. write and evaluate a logarithmic equation to show how many weeks it takes to plant 500 herbal plants. round your answer to the nearest whole number of weeks. use the keypad to enter the answers in the boxes. more symbols can be found using the drop - down arrow at the top of the keypad. a logarithmic equation to find how many weeks it takes to plant 500 herbal plants is . it takes about weeks to plant 500 herbal plants.

Answer

Explanation:

Step1: Set up the equation

We know that $H(x)=500$ and $H(x) = 50\cdot e^{\frac{x}{5}}$. So, $500=50\cdot e^{\frac{x}{5}}$.

Step2: Simplify the equation

Divide both sides of the equation $500 = 50\cdot e^{\frac{x}{5}}$ by 50. We get $\frac{500}{50}=e^{\frac{x}{5}}$, which simplifies to $10 = e^{\frac{x}{5}}$.

Step3: Convert to logarithmic form

The natural - logarithm form of the equation $10 = e^{\frac{x}{5}}$ is $\ln(10)=\frac{x}{5}$.

Step4: Solve for x

Multiply both sides of the equation $\ln(10)=\frac{x}{5}$ by 5. So, $x = 5\ln(10)$.

Step5: Evaluate the value of x

We know that $\ln(10)\approx2.3026$. Then $x = 5\times2.3026=11.513\approx12$.

Answer:

A logarithmic equation to find how many weeks it takes to plant 500 herbal plants is $\ln(10)=\frac{x}{5}$. It takes about 12 weeks to plant 500 herbal plants.