Problem 52 - Calculus and its applications

Question

Solve for \(t\). $$ e^{2 t}=1000 $$

Step-by-Step Solution

Answer

\( t \approx 3.454 \)

Step 1: Understand the Equation

The equation \( e^{2t} = 1000 \) contains an exponential function where the base is the mathematical constant \( e \), approximately 2.718. Our goal is to solve for the variable \( t \).

Step 2: Take the Natural Logarithm

To solve for \( t \), we need to eliminate the exponential part. We do this by taking the natural logarithm (ln) of both sides of the equation. This gives us:\[ \ln(e^{2t}) = \ln(1000) \]

Step 3: Simplify Using Logarithmic Identity

Use the logarithmic identity \( \ln(e^x) = x \), to simplify the left side. So, \( \ln(e^{2t}) = 2t \). Now the equation becomes:\[ 2t = \ln(1000) \]

Step 4: Solve for \( t \)

Isolate \( t \) by dividing both sides of the equation by 2:\[ t = \frac{\ln(1000)}{2} \]

Step 5: Calculate \( t \)

Calculate \( \ln(1000) \). Using a calculator, \( \ln(1000) \approx 6.907755 \,\). Therefore, \[ t = \frac{6.907755}{2} \approx 3.4538775 \]

Key Concepts

The Natural Logarithm

The natural logarithm, often represented by \( \ln \), is a special type of logarithm. Its base is the number \( e \), which is roughly equal to 2.718. Logarithms are used to solve equations where the variable is an exponent, like \( e^{2t} = 1000 \). When you apply the natural logarithm to both sides, you can simplify the equation because of the unique properties of \( \ln \).The property \( \ln(e^x) = x \) is particularly useful. Applying \( \ln \) helps "cancel out" the exponential function, making equations much simpler. In our example, \( \ln(e^{2t}) = 2t \). Understanding this property allows for easier manipulation of equations when solving for unknown variables. In practice, using \( \ln \) is straightforward, just remember you are "undoing" the exponential nature of \( e \). This concept is foundational in calculus and other advanced math fields.

Logarithmic Identity

Logarithmic identities are formulas that help simplify expressions involving logarithms. They are essential for solving many kinds of equations, particularly those involving exponents. One key identity used in the step-by-step solution is \( \ln(e^x) = x \). This identity states that the natural logarithm of \( e \) raised to any power \( x \) simplifies directly to \( x \). It takes advantage of the relationship between the exponential function and its inverse, the logarithm.So, when you see \( \ln(e^{2t}) = \ln(1000) \), you directly simplify the left side using the identity: \( \ln(e^{2t}) \) becomes \( 2t \).Understanding and leveraging logarithmic identities can turn seemingly difficult equations into manageable, solvable problems. As you practice, these identities will become more intuitive, helping expedite the solving process in algebra and beyond.

Solving for Variable t

Solving for variable \( t \) in an equation like \( e^{2t} = 1000 \) involves several clear steps. We aim to isolate \( t \) to find its value. First, use the natural logarithm to "log out" the exponential function, reducing complexity: \( \ln(e^{2t}) = \ln(1000) \), which simplifies to \( 2t = \ln(1000) \) using the identity \( \ln(e^x) = x \).Next, solve the simplified equation. To isolate \( t \), divide each side by 2: \( t = \frac{\ln(1000)}{2} \).Finally, calculate \( \ln(1000) \) using a calculator. The result, \( t = \frac{6.907755}{2} \), gives approximately \( t \approx 3.4538775 \).Breaking it down into these steps makes solving exponential equations straightforward. Recognizing the use of logarithms and identities turns a challenging problem into a step-by-step solution. By understanding these core concepts and practicing more, the process becomes more intuitive and faster over time.