Solve the equation algebraically and check graphically.

Question

e Superscript tequals984
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Part 1
tequals
  
enter your response here
​(Type an integer or decimal rounded to three decimal places as​ needed.)

Answer 1

To solve the equation \( e^t = 984 \) algebraically, we'll follow these steps:

1. Take the natural logarithm of both sides: \[ \ln(e^t) = \ln(984) \]

2. Using the property of logarithms that states \( \ln(e^x) = x \), we simplify the left side: \[ t = \ln(984) \]

3. Now, we'll calculate \( \ln(984) \) using a calculator: \[ \ln(984) \approx 6.891 \]

Thus, the solution for \( t \) is approximately: \[ t \approx 6.891 \]

Checking Graphically:

To check this graphically, you can plot the function \( y = e^t \) and the line \( y = 984 \). The point of intersection will give you the solution for \( t \).

1. Plot the function \( y = e^t \), which is an exponential curve that starts at \( (0, 1) \) and increases rapidly as \( t \) increases.

2. Draw a horizontal line at \( y = 984 \).

3. The \( t \)-coordinate of the intersection point of the curve and the line represents the solution for \( t \).

If you perform this plot using graphing software or a graphing calculator, you should see that the point of intersection aligns with \( t \approx 6.891 \), confirming our algebraic solution.

Hence, the final answer is: \[ t \approx 6.891 \] (rounded to three decimal places).