To solve the equation \( e^t = 183 \) algebraically, we can take the natural logarithm of both sides.
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Start with the given equation: \[ e^t = 183 \]
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Take the natural logarithm of both sides: \[ \ln(e^t) = \ln(183) \]
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Using the property of logarithms that \( \ln(e^x) = x \), we can simplify the left side: \[ t = \ln(183) \]
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Now, calculate \( \ln(183) \) using a calculator: \[ t \approx 5.216 \]
Thus, the solution to the equation \( e^t = 183 \) is: \[ t \approx 5.216 \quad (\text{rounded to three decimal places: } 5.216) \]
Graphical Check
To check this graphically, plot the function \( y = e^t \) and \( y = 183 \). The intersection point of these curves will give the solution for \( t \).
- The curve \( y = e^t \) is an exponential curve, which increases continuously.
- The line \( y = 183 \) is a horizontal line.
You should find that these two curves intersect at approximately \( t \approx 5.216 \), confirming our algebraic solution.
Remember, you can use graphing tools or calculators to visualize the functions better and see where they intersect.