Problem 4 - Algebra and Trigonometry

Question

Find the solution of the exponential equation, correct to four decimal places. $$ e^{3 x}=12 $$

Step-by-Step Solution

Answer

The solution is \( x \approx 0.8283 \).

Step 1: Understand the Equation

The given equation is an exponential equation with base \( e \). We have \( e^{3x} = 12 \). The goal is to solve for \( x \).

Step 2: Apply Natural Logarithm

To isolate \( x \), take the natural logarithm (ln) of both sides of the equation. This helps in leveraging the property that the natural log and the exponential function are inverses.\[ \ln(e^{3x}) = \ln(12) \]

Step 3: Simplify Using Logarithmic Identity

Utilize the identity \( \ln(e^y) = y \) to simplify the left side of the equation. This gives:\[ 3x = \ln(12) \]

Step 4: Solve for x

To solve for \( x \), divide both sides of the equation by 3:\[ x = \frac{\ln(12)}{3} \]

Step 5: Calculate the Value

Now, compute the value using a calculator to find \( \ln(12) \) and divide by 3 to get:\[ x \approx \frac{2.4849}{3} \approx 0.8283 \]Thus, the solution, correct to four decimal places, is \( x \approx 0.8283 \).

Key Concepts

Natural Logarithm

The natural logarithm, often represented as "ln", is a logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. The natural logarithm function is the inverse of the exponential function with base \( e \). This means that if you have \( y = e^x \), then \( x = \ln(y) \).
It's widely used in mathematics, especially in calculus and solving exponential equations, because it simplifies the manipulation of expressions involving growth models or compounding.
When you take the natural logarithm of both sides in an equation like \( e^{3x} = 12 \), it allows you to bring down the exponent. This property is key because it turns a potentially complex exponential equation into a linear one, making it much easier to solve.

Exponential Functions

Exponential functions are mathematical expressions where the variable is in the exponent. A common form is \( f(x) = a^x \), where \( a \) is a positive constant. In the given exercise, the base of the exponential function is the natural base \( e \), which makes it an exponential function \( f(x) = e^{3x} \).
Exponential functions have some distinct characteristics:

• They start slow but grow very fast as \( x \) increases.
• They never touch the x-axis (horizontal asymptote at \( y = 0 \)).
• They are continuously increasing if the base \( a \) is greater than 1.

When solving an equation of the form \( e^{3x} = 12 \), our aim is to determine the value of \( x \) for which the equality holds. We use the properties of logarithms to transform the equation and solve it.

Logarithmic Identities

Logarithmic identities are mathematical rules that simplify the manipulation and understanding of logarithms, thereby helping to solve equations more easily. Some of these identities include:

• \( \log_a(a^x) = x \)
• \( \ln(e^x) = x \)
• Product Identity: \( \log_a(uv) = \log_a(u) + \log_a(v) \)
• Quotient Identity: \( \log_a\left(\frac{u}{v}\right) = \log_a(u) - \log_a(v) \)
• Power Identity: \( \log_a(u^n) = n \cdot \log_a(u) \)

In the context of the given problem, \( \ln(e^{3x}) \) resolves directly to \( 3x \) using the identity \( \ln(e^y) = y \). This simplifies our equation significantly and helps us isolate the variable \( x \) by reducing the exponential expression to a simple algebraic one.