Problem 277 - Calculus Volume 1

Question

Solve the exponential equation exactly. \(e^{3 x}-15=0\)

Step-by-Step Solution

Answer

\(x = \frac{\ln(15)}{3}\)

Step 1: Isolate the Exponential

First, we want to isolate the exponential term. Starting with the equation \( e^{3x} - 15 = 0 \), we add 15 to both sides of the equation. This gives us \( e^{3x} = 15 \).

Step 2: Apply the Natural Logarithm

To solve for \( x \), take the natural logarithm of both sides. This results in \( \ln(e^{3x}) = \ln(15) \). Utilizing the logarithmic identity \( \ln(e^y) = y \), we simplify the left side to obtain \( 3x = \ln(15) \).

Step 3: Solve for x

Finally, solve for \( x \) by dividing both sides by 3, resulting in \( x = \frac{\ln(15)}{3} \).

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \( \ln \), is a useful tool in mathematics for dealing with exponential expressions. In mathematics, the natural logarithm is the inverse of the exponential function when the base is \( e \), where \( e \approx 2.71828 \).
This means:

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

The natural logarithm is particularly powerful because it simplifies calculations involving base \( e \) exponentials. When we apply \( \ln \) on both sides of an equation like \( e^{3x} = 15 \), it helps us "break apart" the exponential and allows easy arithmetic manipulation.
Understanding the natural logarithm is key to solving equations involving exponentials, especially when the exact solution is required.

Solving Equations

Solving equations, especially those involving exponential terms, requires a strategic approach. The first step is often to isolate the term with the variable: here, \( e^{3x} - 15 = 0 \) simplifies to \( e^{3x} = 15 \) by adding 15 to both sides.

• Isolate the variable term to make calculations easier.
• Apply arithmetic operations equally to both sides to maintain the equation's balance.

Once the exponential term \( e^{3x} \) is isolated, we apply the natural logarithm to assist in removing the exponential expression.
This simplifies the problem significantly by changing it from a form involving exponents to a linear form, such as \( 3x = \ln(15) \), allowing us to more easily solve for \( x \) through basic algebraic steps like division.
Solving such equations often boils down to a series of strategic simplifications making the process systematic and approachable.

Logarithmic Identities

Logarithmic identities are important mathematical rules that help simplify complex expressions, especially in solving equations like exponentials. A critical identity, \( \ln(e^y) = y \), allows us to extract the exponent when applying a natural logarithm to an exponential expression.

• \( \ln(e^x) = x \), simply stripping away the exponential layer.
• \( \ln(ab) = \ln(a) + \ln(b) \), breaking down products.
• \( \ln(a^n) = n \ln(a) \), simplifying power terms.

These properties are especially useful in mathematical simplifications involving logarithms. For instance, in \( \ln(e^{3x}) \), the identity helps to immediately simplify the expression to \( 3x \), unveiling the path to solving the equation.
Recognizing and applying these identities allows mathematicians and students alike to tackle mathematical challenges effortlessly by simplifying and reducing equations to their most manageable forms.