Problem 73 - College Algebra with Modeling and Visualization

Question

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$5 e^{x}+2=20$$

Step-by-Step Solution

Answer

The solution is approximately \( x \approx 1.28 \).

Step 1: Isolate the exponential term

Begin by isolating the term involving the exponential function on one side of the equation. The original equation is: \( 5e^x + 2 = 20 \). Subtract 2 from both sides to get: \( 5e^x = 18 \).

Step 2: Solve for the exponential expression

Next, to isolate \( e^x \), divide both sides of the equation by 5. This gives: \( e^x = \frac{18}{5} \).

Step 3: Apply the natural logarithm

Since \( e^x = \frac{18}{5} \), take the natural logarithm (ln) of both sides to solve for \( x \). This results in: \( x = \ln\left(\frac{18}{5}\right) \).

Step 4: Use a calculator to approximate the answer

Using a calculator, approximate \( \ln\left(\frac{18}{5}\right) \) to the nearest hundredth. The result is approximately \( x \approx 1.28 \).

Key Concepts

Change of Base Formula

In mathematics, we often encounter logarithmic equations that require us to switch from one base to another. The change of base formula is a handy tool that makes it possible to convert a logarithm of any base to another base that is more convenient to use. This formula is particularly useful when dealing with calculators, which typically only perform logarithms in base 10 (common logarithm) or base e (natural logarithm).

The change of base formula is written as follows:

• For a logarithm in base 'b', it can be expressed as: \[\log_b{a} = \frac{\log_c{a}}{\log_c{b}}\]where 'c' is the new base that you convert to.

In our exercise, we didn't directly use this formula, but understanding it helps you solve more complex logarithmic equations. When solving exponential equations, like our example, the natural logarithm often becomes the preferred choice since it simplifies dealing with equations involving the mathematical constant e.

Natural Logarithm

The natural logarithm is a special logarithm with the base of e, a transcendental number approximately equal to 2.71828. Notation for natural logarithm is expressed as \( \ln \).

In the given equation, once we isolated the exponential function as \( e^x = \frac{18}{5} \), we applied the natural logarithm to both sides. This operation is key in unlocking the value of \( x \) because the property of logarithms tells us that \( \ln(e^x) = x \cdot \ln(e) = x \).

• This simplifies the solution significantly, because \( \ln(e) \) is 1, thus leaving us with just \( x \).
• The step with \( \ln \) operation is crucial since it transforms an equation that's otherwise difficult to solve into a simple algebraic form.

Natural logarithms are integral in calculus and are often encountered in problems involving exponential growth and decay. They allow us to work with exponential expressions in a meaningful way that connects well with geometric growth patterns.

Approximation

In mathematics, approximation helps us gain insights into numerical values that are not exactly solvable. This is especially useful with exponential equations, where exact values of logarithmic terms might be cumbersome.

In our exercise, after applying the natural logarithm, we approximated \( \ln\left(\frac{18}{5}\right) \) to the nearest hundredth.
To ensure accuracy:

• Always express your answer to at least two decimal places when required, unless specified otherwise.
• Make use of tools like calculators that provide precise logarithmic values.

This rounded answer, \( x \approx 1.28 \), allows us to maintain a balance between precision and simplicity, critical for real-world problem-solving scenarios where exactness may not always be necessary but an accurate estimation is desirable.