Problem 15 - Precalculus

Question

In Exercises \(1-33,\) solve the equation analytically. $$ 2000 e^{0.1 t}=4000 $$

Step-by-Step Solution

Answer

The solution is approximately \( t \approx 6.93 \).

Step 1: Isolate the Exponential Term

Begin by dividing both sides of the equation by 2000 to isolate the exponential term. This gives us: \[ e^{0.1t} = \frac{4000}{2000} \] Simplifying the right side, we find: \[ e^{0.1t} = 2 \]

Step 2: Apply the Natural Logarithm

To remove the exponential, take the natural logarithm (\( \ln \)) of both sides of the equation: \[ \ln(e^{0.1t}) = \ln(2) \] Since \( \ln(e^x) = x \), this simplifies to: \[ 0.1t = \ln(2) \]

Step 3: Solve for t

Finally, solve for \( t \) by dividing both sides by 0.1: \[ t = \frac{\ln(2)}{0.1} \] Calculate the value: \[ t \approx \frac{0.693}{0.1} = 6.93 \]

Key Concepts

Understanding Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In simpler terms, this means the exponent is what varies. For example, in the function \( y = a^x \), \( a \) is a constant (besides being a positive number), and \( x \) is a variable. Exponential functions grow rapidly because the variable \( x \) is in the exponent.
When solving equations with exponential functions, you often look to isolate the exponential term on one side of the equation. By doing so, you make it easier to apply operations that help solve for the variable. In the original equation \( 2000e^{0.1t} = 4000 \), isolating the exponential means dividing both sides by 2000 to get \( e^{0.1t} = 2 \). This step is crucial as it sets the stage for using other mathematical tools to solve for the exponent.

Introduction to Natural Logarithms

Natural logarithms are a specific type of logarithms that have the base of \( e \), which is approximately equal to 2.71828. The purpose of logarithms is to figure out what power (or exponent) you must raise a given base to obtain a specific number. Natural logarithms are denoted by \( \ln \).
In this exercise, when you have \( e^{0.1t} = 2 \), taking the natural logarithm of both sides helps "undo" the exponential function. So, \( \ln(e^{0.1t}) = \ln(2) \). Thanks to the property \( \ln(e^x) = x \), you simplify \( \ln(e^{0.1t}) \) to \( 0.1t \). This process is vital because it transforms the equation into a linear format, making it easier to solve for \( t \).
Through natural logarithms, you can effectively work through exponential equations by reducing them to simple linear equations.

Using Analytical Methods to Solve Equations

Analytical methods refer to the precise, step-by-step approach used to find a solution to an equation. These methods follow logical progressions and calculations without the need for graphical representations or numerical estimation.
In solving the equation \( 2000e^{0.1t} = 4000 \) analytically, we employed both algebraic manipulations and logarithmic identities. Initially, the first step involved isolating \( e^{0.1t} \) by dividing. This made it straightforward to apply a natural logarithm to both sides and eliminate the exponential exponent, transforming the equation into \( 0.1t = \ln(2) \).
The final step in this analytical method was to solve for \( t \) using basic arithmetic operations, namely division. By dividing both sides by 0.1, the solution is expressed as \( t = \frac{\ln(2)}{0.1} \). Calculating this expression gives a precise value of approximately 6.93. Analytical methods offer a clear, systematic means to solve equations, particularly when precision is essential.