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Chapter 9: Problem 19

Solve. Where appropriate, include approximations to three decimal places. $$ e^{t}=50 $$

Short Answer

Expert verified
t \approx 3.912

Step by step solution

01

- Understand the Equation

The equation given is an exponential equation where the base is the mathematical constant \(e\) and the exponent is \(t\). The equation is \(e^{t} = 50\).
02

- Apply Natural Logarithm

To solve for \(t\), take the natural logarithm (ln) on both sides of the equation: \( \text{ln}(e^{t}) = \text{ln}(50) \).
03

- Use Logarithm Property

Utilize the property of logarithms that states \( \text{ln}(e^{t}) = t \text{ln}(e)\). Since \( \text{ln}(e) = 1 \), the equation simplifies to \( t = \text{ln}(50) \).
04

- Calculate the Natural Logarithm

Compute the natural logarithm of 50 using a calculator: \( t = \text{ln}(50) \approx 3.912 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithm
The natural logarithm, often written as \( \text{ln} \), is a type of logarithm where the base is the mathematical constant \( e \) (approximately 2.718). The \( \text{ln} \) function helps us solve equations where the unknown is an exponent with base \( e \). When you see an equation like \( e^{t} = 50 \), you can use the natural logarithm to 'bring down' the exponent.

To solve an exponential equation using the natural logarithm, follow these steps:
  • Take the natural logarithm of both sides of the equation.
  • The natural logarithm has the property: \( \text{ln}(e^{x}) = x \), which simplifies the equation.
In this way, the exponent becomes more manageable, allowing you to solve for the unknown variable.
Logarithmic Properties
Logarithms have several properties that come in handy when solving equations. One important property is \( \text{ln}(e^{x}) = x \text{ln}(e) \). Given that \( \text{ln}(e) = 1 \), this simplifies to \( \text{ln}(e^{x}) = x \).

In our example, we have \( e^{t} = 50 \). By applying the natural logarithm to both sides, we get \( \text{ln}(e^{t}) = \text{ln}(50) \).

Using the property mentioned earlier, we can simplify \( \text{ln}(e^{t}) \) to \( t \text{ln}(e) \). Since \( \text{ln}(e) = 1 \), the equation reduces to \( t = \text{ln}(50) \).
  • This approach helps us turn complex exponential equations into simpler linear forms.
  • Knowing these logarithmic properties can save time and reduce errors in solving equations.
Approximations
Sometimes, exact values are challenging to compute or represent. This is where approximations come in handy. Approximations allow us to express solutions in a simpler form, especially for practical applications. In this problem, after taking the natural logarithm, we get \( t = \text{ln}(50) \).

We use a calculator to find that \( \text{ln}(50) \approx 3.912 \).

Approximations are often rounded to a specific number of decimal places, depending on the required accuracy. Here, we rounded \( t \) to three decimal places.
  • Approximations make complex solutions easier to understand and use.
  • Always state the accuracy level (e.g., three decimal places) for clarity.
Remember, while approximations are useful, they are not exact, so be mindful of the context in which you use them.

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