Problem 20 - Precalculus

Question

Solve the exponential equations. Make sure to isolate the base to a power first. Round our answers to three decimal places. $$21-4 e^{0.1 x}=5$$

Step-by-Step Solution

Answer

The solution is approximately \( x = 13.860 \).

Step 1: Isolate the Exponential Term

To isolate the term involving the exponential, first move all terms not involving the exponential to the other side. We start with the equation: \[ 21 - 4e^{0.1x} = 5 \] Subtract 21 from both sides: \[ -4e^{0.1x} = 5 - 21 \] Simplify the right side: \[ -4e^{0.1x} = -16 \]

Step 2: Solve for the Exponential

Now that the exponential term is isolated, divide both sides by -4 to solve for the exponential expression: \[ e^{0.1x} = \frac{-16}{-4} \] Simplify the right side: \[ e^{0.1x} = 4 \]

Step 3: Apply Natural Logarithm

To solve for the variable \( x \), take the natural logarithm of both sides to eliminate the exponential: \[ \ln(e^{0.1x}) = \ln(4) \] Using the power rule for logarithms, simplify the left side: \[ 0.1x \ln(e) = \ln(4) \] Since \( \ln(e) = 1 \), we have: \[ 0.1x = \ln(4) \]

Step 4: Solve for x

To find \( x \), divide both sides by 0.1: \[ x = \frac{\ln(4)}{0.1} \] Calculate the value of \( \ln(4) \) and then perform the division: \[ x \approx \frac{1.386}{0.1} \] Thus, \( x \approx 13.860 \) after rounding to three decimal places.

Key Concepts

Isolating the Base

Before solving an exponential equation, it's crucial to first isolate the base that is raised to a power. In the given equation, \( 21 - 4e^{0.1x} = 5 \), the term involving the exponential must be singled out on one side of the equation. This gives us a simplified view, preparing the equation for its subsequent steps. Here's how you can achieve that:

• Identify the term containing the exponential expression, which is \( -4e^{0.1x} \).
• To isolate this term, start by moving the constant term on the same side of the equation to the other side. In our example, subtract 21 from both sides, resulting in \( -4e^{0.1x} = -16 \).
• After the operation, we should eliminate any multipliers of the exponential term. Divide both sides by \(-4\), leading to the isolated form \( e^{0.1x} = 4 \).

Isolation is crucial because it sets the stage for applying logarithmic methods to solve for the variable.

Natural Logarithm

The natural logarithm is a powerful tool used to solve exponential equations. It's denoted as \( \ln \) and is specifically the logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.718. When you face an equation with an isolated exponential base, taking the natural logarithm of both sides can help you solve for unknowns like \( x \). In our setup, we have \( e^{0.1x} = 4 \), and we take the natural log of both sides to proceed:

• The expression becomes \( \ln(e^{0.1x}) = \ln(4) \).
• Applying the natural log helps to bring the exponent down and allows us to manipulate equations in a linear form. This is crucial for simplifying exponential equations.

Using the natural logarithm simplifies complex exponential expressions and makes it possible to solve for variables easily.

Power Rule for Logarithms

The power rule for logarithms is another critical concept when solving exponential equations. This rule states that for any positive number \( a \) and \( n \), the expression \( \log(a^n) \) can be re-written as \( n \log(a) \). In terms of natural logarithms, this means:

• The expression \( \ln(e^{0.1x}) \) becomes \( 0.1x \ln(e) \).
• In the context of our example, we take advantage of the fact that \( \ln(e) = 1 \).
• This simplifies to \( 0.1x = \ln(4) \), allowing you to easily isolate \( x \) by dividing both sides by 0.1.

The power rule for logarithms enables you to transform complicated exponentials into straightforward linear equations, which are much easier to handle computationally.