Question

Express y as a function of \(x .\) The constant \(C\) is a positive number. \(\ln y=-2 x+\ln C\)

Short answer

The function is \(y = C \cdot e^{-2x}\).

Step-by-step solution

Move the Natural Logarithm to One Side

The given equation is \(\backslash ln y = -2x + \backslash ln C\). Let's move \(\backslash ln y\) to one side to isolate the logarithmic part.

Exponentiate Both Sides to Eliminate Natural Logarithm

To get rid of the natural logarithm, exponentiate both sides of the equation: \[ e^{\backslash ln y} = e^{-2x + \backslash ln C} \]

Apply Exponentiation Properties

Apply the property of exponents \[e^{a + b} = e^{a} \cdot e^{b}\backslash\]: \[ e^{\backslash ln y} = e^{-2x} \cdot e^{\backslash ln C} \]

Simplify the Equation

The term \(e^{\backslash ln y}\) simplifies to \(y\), and \(e^{\backslash ln C}\) simplifies to \(C\) because the exponential and natural logarithm functions are inverse operations. Now the equation is: \[ y = e^{-2x} \cdot C \]

Write the Final Function

Rearrange the expression to clearly show \(y\) as a function of \(x\): \[ y = C \cdot e^{-2x} \]

Key concepts

Natural Logarithm

A natural logarithm, denoted as \(\backslash ln\), is a logarithm with the base \(e\). The constant \(e\) is approximately equal to 2.71828 and is an important mathematical constant. Natural logarithms are used to solve equations involving exponential growth and decay. For example, \(\backslash ln y\) represents the exponent to which \(e\) must be raised to equal \(y\). This function is incredibly useful in many fields, such as biology, economics, and physics.

When you encounter \(\backslash ln y = -2x + \backslash ln C\), remember that this is showing the relationship between the variables under the natural logarithm.

Exponentiation

Exponentiation is a mathematical operation involving a base raised to a power. In the context of natural logarithms, exponentiation can be used to reverse the logarithm effect. For example, in the equation \(\backslash ln y = -2x + \backslash ln C\), we can exponentiate both sides by the base \(e\) to get rid of the natural logarithm: \(\backslash e^{\backslash ln y} = \backslash e^{-2x + \backslash ln C}\).

This simplifies to \(y = \backslash e^{-2x} \backslash \times \backslash e^{\backslash ln C}\). By exponentiating, we are using the property that \( \backslash e^{\backslash ln a} = a\), which helps isolate \(y\) in terms of \(x\). This is a fundamental skill in algebra and calculus.

Inverse Operations

Inverse operations are pairs of operations that reverse the effect of each other. For instance, addition and subtraction are inverse operations, as are multiplication and division. In our context, the natural logarithm (\(\backslash ln\)) and the exponential function (\(\backslash e^x\)) are inverse operations.

This means that when you apply the exponential function to a natural logarithm, they cancel out: \(\backslash e^{\backslash ln y} = y\). This property is crucial for solving the equation \( \backslash ln y = -2x + \backslash ln C \), as exponentiating both sides allows us to eliminate the \( \backslash ln\) and express \(y\) in terms of \(x\). Understanding inverse operations is essential for solving many algebraic and calculus problems.

Properties of Exponents

Properties of exponents are mathematical rules that simplify operations involving exponential functions. Some important properties include:

• \(a^{m} \backslash times a^{n} = a^{m+n}\): Multiplying exponents with the same base.
• \( (a^{m})^{n} = a^{mn} \): Raising a power to another power.
• \( a^{0} = 1 \): Anything raised to the power of zero.
• \( a^{-n} = \frac{1}{a^{n}} \): Negative exponents represent the reciprocal.

In our problem, we used the property \( \backslash e^{a+b} = \backslash e^{a} \backslash times \backslash e^{b} \) to break down \( \backslash e^{-2x+\backslash ln C} \) into \( \backslash e^{-2x} \backslash times \backslash e^{\backslash ln C} \). This allowed us to simplify the expression further: \( y = C \backslash times \backslash e^{-2x} \). Efficiently using these properties is key to simplifying and solving exponential equations.