Problem 6 - Algebra and Trigonometry

Question

Find the solution of the exponential equation, rounded to four decimal places. \(e^{3 x}=12\)

Step-by-Step Solution

Answer

The solution is approximately \(x = 0.8283\).

Step 1 - Understand the Equation

We start with the given equation \(e^{3x} = 12\). The goal is to isolate \(x\) by transforming the equation using logarithms.

Step 2 - Apply the Natural Logarithm

Take the natural logarithm (\(\ln\)) of both sides of the equation to remove the exponential. This gives us \(\ln(e^{3x}) = \ln(12)\).

Step 3 - Simplify Using Logarithm Rules

Use the property of logarithms that \(\ln(e^y) = y\). Therefore, \(\ln(e^{3x}) = 3x\). The equation simplifies to \(3x = \ln(12)\).

Step 4 - Solve for x

Divide both sides of the equation by 3 to solve for \(x\). This gives \(x = \frac{\ln(12)}{3}\).

Step 5 - Calculate the Value

Use a calculator to find \(\ln(12)\), which is approximately 2.4849. Then calculate \(x = \frac{2.4849}{3} \approx 0.8283\).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a mathematical function that is used to solve equations involving exponential terms, particularly those with base \( e \), the Euler's number. \( e \) is approximately equal to 2.71828 and is an important constant in mathematics, especially in calculus and exponential growth models.

The natural logarithm is the inverse operation of exponentiation with base \( e \). This means that if you exponentiate a number \( x \) to get \( e^x \), applying the natural logarithm retrieves the original \( x \). For example, if \( e^x = y \), then \( \ln(y) = x \).

When working with equations like \( e^{3x} = 12 \), taking the natural logarithm on both sides helps to "undo" the exponential, allowing you to solve for the variable by bringing it down from the exponent position.

Properties of Logarithms

Logarithms have several important properties that we regularly use when solving equations. These properties make the process of manipulating equations easier, especially those involving exponents.

One key property is that \( \ln(e^y) = y \). This property allows us to simplify expressions where the logarithm of an exponential term is involved. For example, in the equation \( \ln(e^{3x}) \), you can directly simplify this to \( 3x \).

Moreover, logarithms are useful in simplifying product and power expressions. Some of these properties include:

• The product rule: \( \ln(a \cdot b) = \ln(a) + \ln(b) \)
• The quotient rule: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)
• The power rule: \( \ln(a^b) = b \cdot \ln(a) \)

These rules help consolidate complex logarithmic expressions, making solving equations much more straightforward.

Solving Equations

Solving equations, particularly exponential equations, requires a systematic approach to isolate the desired variable. In our example, the equation \( e^{3x} = 12 \) is an exponential equation where the variable is in the exponent.

The process typically involves:

• Applying logarithms to remove the exponential term. For example, we used \( \ln \) to handle the equation.
• Simplifying the equation using properties of logarithms.
• Performing algebraic operations to isolate the variable. In this case, dividing both sides by 3 once the logarithm has been applied.
• Using a calculator to find approximate numerical values, which is especially useful when dealing with constants such as \( \ln(12) \).

By carefully applying these steps, you can solve exponential equations efficiently, arriving at solutions that can be rounded to the required precision, such as four decimal places.