Problem 16 - Algebra for College Students

Question

Solve each exponential equation and express approximate solutions to the nearest hundredth. $$ 2 e^{x}=12.4 $$

Step-by-Step Solution

Answer

The solution is approximately \( x = 1.82 \).

Step 1: Isolate the Exponential Term

The given equation is \( 2e^{x} = 12.4 \). First, divide both sides by 2 to isolate the exponential term: \[ewecommand{\divide}{\frac{12.4}{2}}\ e^{x} = \divide = 6.2 \]

Step 2: Apply the Natural Logarithm

To solve for \( x \), take the natural logarithm (ln) of both sides: \[ \ln(e^{x}) = \ln(6.2) \]

Step 3: Use the Logarithmic Identity

Use the identity \( \ln(e^{x}) = x \) to simplify the left-hand side of the equation:\[ x = \ln(6.2) \]

Step 4: Calculate the Natural Logarithm

Using a calculator, find the natural logarithm of 6.2:\[ x \approx 1.824 \]

Step 5: Round to the Nearest Hundredth

Round the approximate solution to the nearest hundredth:\[ x \approx 1.82 \]

Key Concepts

Natural Logarithm

The natural logarithm is a mathematical function denoted as \( \ln(x) \). It is the logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. The natural logarithm is particularly useful in calculus and when dealing with exponential equations because it simplifies equations involving exponentials.

• \( \ln(e) = 1 \) because \( e^1 = e \)
• \( \ln(1) = 0 \) because \( e^0 = 1 \)

The function \( \ln(x) \) is the inverse of the exponential function \( e^x \). This means that for any number \( x \), if you take \( e^x \) and then apply \( \ln \), you will get back to the original number \( x \). This relationship is pivotal when solving exponential equations as it provides a way to "undo" the exponential operation.

Solve Exponential Equations

Solving exponential equations often involves transforming the equation so that the unknown variable is no longer in the exponent. For the equation \( 2e^x = 12.4 \):

• First, isolate the exponential term by dividing both sides by 2, resulting in \( e^x = 6.2 \).
• Then, take the natural logarithm of both sides: \( \ln(e^x) = \ln(6.2) \).

At this point, the equation has been converted to involve a logarithm, which can downgrade the power of x from an exponent to linear form, making it easier to handle and solve. It essentially "flattens" the exponential curve, providing a clear path to finding the variable.

Logarithmic Identity

A critical tool in solving exponential equations is understanding and using logarithmic identities. These identities can simplify complex expressions and are key when working with both exponential and logarithmic equations.

One of the most important identities used in solving exponential equations is \( \ln(e^x) = x \). This identity is derived from the relationship between exponentials and logarithms, where the natural logarithm is the inverse operation of exponentiation.

When you apply this identity to \( \ln(e^x) \), the \( x \) is effectively "brought down" from the exponent, rendering an equation linear, which is beautifully simple to solve. This identity allowed us to simplify \( \ln(e^x) = \ln(6.2) \) to just \( x = \ln(6.2) \) in the step-by-step solution, thereby revealing the answer almost immediately. Familiarity with such identities transforms our problem-solving toolkit, creating shortcuts and solutions that are both efficient and understandable.