Problem 44 - Algebra 2

Question

Solve each equation. Round to the nearest ten-thousandth. $$ e^{3 x}=4 $$

Step-by-Step Solution

Answer

The solution is approximately \( x = 0.4621 \).

Step 1: Understand the problem

We are given the equation \( e^{3x} = 4 \) and are tasked with solving for \( x \). The expression involves the natural exponential function \( e \). Our goal is to isolate \( x \) in the equation.

Step 2: Apply the natural logarithm

To isolate \( x \), we take the natural logarithm (ln) of both sides of the equation. This gives us:\[ \ln(e^{3x}) = \ln(4) \]This step uses the property that \( \ln(e^y) = y \).

Step 3: Simplify using logarithmic identities

Using the property \( \ln(e^{3x}) = 3x \), we simplify the left side of the equation to obtain:\[ 3x = \ln(4) \]

Step 4: Solve for x

To solve for \( x \), divide both sides of the equation by 3:\[ x = \frac{\ln(4)}{3} \]Calculate \( \ln(4) \) using a calculator, and then divide by 3 to find \( x \).

Step 5: Calculate and round the result

First, find \( \ln(4) \approx 1.386294361 \). Then, divide by 3 to find \( x \approx 0.46209812 \).Round this result to the nearest ten-thousandth to get \( x \approx 0.4621 \).

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \( \ln(x) \), is a logarithm with the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. The natural logarithm is the inverse of the exponential function.

• Inverse Relationship: This means if \( e^y = x \), then \( \ln(x) = y \).
• Key Property: An important property used in calculations is \( \ln(e^y) = y \). This identity plays a crucial role in simplifying exponential equations, as seen in the original exercise.

By applying the natural logarithm to both sides of an equation like \( e^{3x} = 4 \), we employ its inverse nature to simplify and solve for variables. This simplification is a fundamental step in tackling exponential equations.

Rounding Numbers

Rounding numbers is the process of adjusting the digits of a number to make it simpler, keeping it as close as possible to the original. In mathematics, rounding is used to attain a specified level of precision, which is particularly useful for approximating irrational numbers or when an exact value is unnecessary.

• Precision: The problem requires rounding the result to the nearest ten-thousandth, which has four decimal places.
• Rules: Generally, if the digit following the desired precision level is 5 or higher, we round up. If not, we round down.

For example, when we calculate \( x \approx 0.46209812 \), the digit in the fifth place is 9, so we round \( 0.46209812 \) to \( 0.4621 \), achieving the necessary precision.

Logarithmic Identities

Logarithmic identities are mathematical rules that make it easier to work with logarithms. They simplify expressions and provide a smooth transition from exponential to logarithmic forms. Here are a few important logarithmic identities:

• Power Rule: \( \ln(a^b) = b \cdot \ln(a) \). This rule is instrumental when dealing with powers inside a logarithm.
• Product Rule: \( \ln(a \cdot b) = \ln(a) + \ln(b) \).
• Quotient Rule: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).

In our example, the Power Rule helps us rewrite and simplify the expression \( \ln(e^{3x}) \) to \( 3x \), because \( \ln(e^y) = y \). These identities assist in breaking down complex expressions into manageable parts, making calculations more straightforward.