Question

Eliminate the parameter in the given parametric equations. \(x=e^{t}, \quad y=e^{3 t}-3\)

Short answer

The cartesian equation is \(y = x^3 - 3\).

Step-by-step solution

Express the parameter in terms of x

Given the parametric equation for x, which is \(x = e^t\), solve for the parameter \(t\). Start by taking the natural logarithm of both sides: \( t = \ln(x) \).

Substitute the parameter in the second equation

Now that we have \(t = \ln(x)\), substitute \(t\) in the second parametric equation \(y = e^{3t} - 3\). Replace \(t\) with \(\ln(x)\) to get: \[ y = e^{3(\ln(x))} - 3 \].

Simplify the expression

Simplify the equation \( y = e^{3(\ln(x))} - 3 \) using the properties of exponents. Recognizing that \(e^{3 \ln(x)} = (e^{\ln(x)})^3 = x^3\), we get: \[ y = x^3 - 3 \].

Key concepts

Eliminating the Parameter

Parametric equations involve a parameter, typically denoted as \( t \), that describes a set of equations with variables \( x \) and \( y \). The goal is usually to eliminate this parameter to express \( y \) solely in terms of \( x \). This often helps in simplifying the analysis or graphing of these equations.

To eliminate the parameter, you begin by solving one of the parametric equations for \( t \). In this case, the equation is \( x = e^t \). To solve for \( t \), you need to "undo" the exponential, which involves using the natural logarithm as it is the inverse function of the exponential.

After expressing \( t \) in terms of \( x \) by using logarithms, substitute this expression into the other equation. This removal of the parameter creates a direct relationship between \( x \) and \( y \), resulting in a single equation devoid of \( t \). This technique simplifies the original parametric form, aiding in a clearer understanding and manipulation of the equation.

Natural Logarithm

The natural logarithm, denoted \( \ln(x) \), is the inverse function of the exponential function \( e^x \). Its primary role is to help solve equations in which the variable is within an exponent. When you apply a natural logarithm to both sides of the equation \( x = e^t \), you get \( t = \ln(x) \).

This step is critical because it translates the parametric form into a simpler algebraic form that no longer contains the parameter \( t \). Here are some key properties of the natural logarithm that are often used:

• \( \ln(e^x) = x \): This property confirms that the natural logarithm effectively reverses an exponential function.
• \( e^{\ln(x)} = x \): This indicates that putting \( e \) and \( \ln \) together cancels out, returning the original value \( x \).

By understanding and applying the natural logarithm, you can manipulate equations to eliminate parameters efficiently.

Properties of Exponents

Exponents are a mathematical shorthand for expressing repeated multiplication. The properties of exponents are key in simplifying complex expressions, especially when dealing with parametric equations like those in this exercise. When you have \( y = e^{3t} - 3 \) and you replace \( t \) with \( \ln(x) \), it becomes \( y = e^{3(\ln(x))} - 3 \).

Using a crucial property of exponents, where \( (e^{\ln(x)})^3 = x^3 \), you can transform the expression into a much simpler form. What happens here is known as the power of a power property, which states that \( (e^a)^b = e^{a \cdot b} \), and when \( a = \ln(x) \), it aligns perfectly with the logarithm rule where \( e^{\ln(x)} = x \).

The specific steps involved in these transformations rely heavily on these exponent rules:

• Power of a Power: \( (a^m)^n = a^{m \cdot n} \)
• Exponent of a Logarithm: \( a^{\log_a(b)} = b \)

Grasping the properties of exponents aids in eliminating parameters and achieving a simplified \( y = x^3 - 3 \), making complex equations manageable and clearer.