To solve the equation \( e^t = 803 \) algebraically, we can take the natural logarithm (ln) of both sides. The natural logarithm and the exponential function are inverses of each other.
Starting with the equation:
\[ e^t = 803 \]
Now, we take the natural logarithm of both sides:
\[ \ln(e^t) = \ln(803) \]
Using the property of logarithms that states \( \ln(e^x) = x \), we get:
\[ t = \ln(803) \]
Now, we can calculate \( \ln(803) \) using a calculator:
\[ t \approx \ln(803) \approx 6.684 \]
So the solution, rounded to three decimal places, is:
\[ t \approx 6.684 \]
Graphical Check
To check graphically, we can plot the functions \( y = e^t \) and \( y = 803 \). The point where these two graphs intersect is the value of \( t \).
- The graph of \( y = e^t \) is an exponential curve that increases rapidly.
- The line \( y = 803 \) is a horizontal line.
You can use graphing software or tools (like Desmos, GeoGebra, or a graphing calculator) to visualize these functions and identify their intersection.
When you graph these functions, the intersection point will occur at approximately \( t \approx 6.684 \), affirming our algebraic solution.
Thus, the final answer is:
\[ t \approx 6.684 \]
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