Problem 8 - Algebra and Trigonometry

Question

Find the solution of the exponential equation, correct to four decimal places. $$ 2 e^{12 x}=17 $$

Step-by-Step Solution

Answer

x ≈ 0.1783

Step 1: Isolate the Exponential Term

Divide both sides of the equation by 2 to isolate the exponential term. \[\frac{2 e^{12x}}{2} = \frac{17}{2} \]This simplifies to:\[ e^{12x} = 8.5 \]

Step 2: Apply the Natural Logarithm

Take the natural logarithm of both sides to eliminate the exponential function. Remember that \( e \) and \( \ln \) are inverse functions.\[\ln(e^{12x}) = \ln(8.5)\]This simplifies to:\[ 12x = \ln(8.5) \]

Step 3: Solve for x

Divide both sides of the equation by 12 to solve for \( x \).\[x = \frac{\ln(8.5)}{12}\]Calculate \( \ln(8.5) \) and divide by 12 to find the value of \( x \).\[ x \approx \frac{2.1401}{12} \approx 0.1783 \]

Key Concepts

Understanding the Natural Logarithm

The natural logarithm is an essential mathematical concept often encountered when dealing with exponential equations. It is denoted by \( \ln \) and primarily concerns the constant \( e \), where \( e \approx 2.71828 \). The natural logarithm is particularly useful for simplifying expressions involving exponents, especially when the base of the exponent is \( e \).

• In the context of solving exponential equations like \( e^{12x} = 8.5 \), applying the natural logarithm to both sides helps you "bring down" the exponent.
• This is because one of the fundamental properties of logarithms states that \( \ln(e^y) = y \). This property exploits the fact that \( e \) and \( \ln \) are inverse functions.

By converting an exponential equation into a logarithmic one, you make it linear and subsequently easier to solve. This is often a key step in isolating the variable of interest in exponential problems.

Inverse Functions and Exponential Equations

Inverse functions play a crucial role in solving equations, including exponential ones. When we talk about inverse functions, we are discussing functions that reverse the effect of each other. If \( f(x) \) is the original function, then its inverse \( f^{-1}(x) \) is such that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

• In the realm of exponential equations, the exponential function \( e^x \) and its inverse, the natural logarithm \( \ln(x) \), are frequently employed together.
• This relationship is pivotal when simplifying solutions because applying \( \ln \) to \( e^{12x} \) leads directly to \( 12x \), effectively "undoing" the exponential operation.

Recognizing this inverse relationship is vital for efficiently transitioning between different forms of equations. It simplifies the process of moving from complex exponential forms to straightforward linear expressions that can be manipulated readily to find solutions.

Simplifying Equations for Solution

Simplifying an equation involves using mathematical operations to isolate the variable you're solving for. It's about breaking down an equation into its simplest form, making it more manageable and solvable. When looking at exponential equations, simplifying might involve several steps, as seen in the original exercise.

• The first step is often isolating the exponential term by dividing or multiplying both sides of the equation. In our example, we divided by 2 to isolate \( e^{12x} \).
• Once isolated, logarithmic tools like the natural logarithm can further simplify the equation by reducing the exponentiation to multiplication.

The last step in this process usually involves basic algebra like division or multiplication to finalize the solution, as demonstrated in dividing both sides by 12. Simplifying an equation step-by-step ensures accuracy and clarity, which is crucial when you're solving for specific values, such as in our example where \( x \approx 0.1783 \).