Problem 32 - Algebra 2

Question

Write an equivalent exponential or logarithmic equation. \(\ln e^{x}=3\)

Step-by-Step Solution

Answer

Equivalent equation: \( e^{x} = e^{3} \).

Step 1: Understand the given equation

The given equation is \( \ln e^{x} = 3 \). Here, the natural logarithm function \( \ln \) is applied to \( e^{x} \), and it equals 3.

Step 2: Use the property of logarithms

Recall the property \( \ln a = b \) means \( a = e^{b} \). For the equation \( \ln e^{x} = 3 \), this property tells us that the expression inside the logarithm equals the exponential of the other side: \( e^{x} = e^{3} \).

Step 3: Simplify the equation

The equation \( e^{x} = e^{3} \) is already in its simplest exponential form. This is the equivalent exponential equation for the given logarithmic equation.

Key Concepts

Exponential Equations

Exponential equations involve expressions where variables are located in the exponent. For example, in the equation \(e^x = e^3\), \(x\) is the exponent. The goal is typically to solve for the variable by isolating it. These types of equations often appear in many fields such as finance, physics, and biology, as they can describe growth and decay processes.

• Exponential equations often have bases like 10 or \(e\), the latter being the natural exponential base.
• The general form looks like \(a^{f(x)} = c\), where you solve for \(x\).

To solve exponential equations, you'll often need to take advantage of transformation techniques like applying logarithms to both sides to "bring down" the exponent.

Properties of Logarithms

Understanding the properties of logarithms is crucial for manipulating and solving logarithmic and exponential equations. A key property used frequently is the natural logarithm property:

• \(\ln a = b\) implies \(a = e^b\).
• This property helps convert a logarithmic equation into an exponential one, thereby simplifying the process of solving it.

Another essential property is the power rule, which states that \(\ln (a^b) = b \cdot \ln a\). This is especially helpful when dealing with equations like \(\ln e^x\), where the \(x\) can be moved in front of the logarithm due to this rule. Understanding these properties allows you to simplify and solve complex logarithmic equations effectively.

Natural Logarithm

The natural logarithm, represented as \(\ln\), is a logarithm to the base \(e\). The number \(e\) is approximately 2.71828 and is a fundamental constant in mathematics.

• \(\ln\) is specific to exponential growth processes like compounding interest and population growth.
• It's used to "undo" the exponential function \(e^x\), as \(\ln(e^x) = x\).

In the original equation \(\ln e^x = 3\), the natural logarithm helps us solve for \(x\) by recognizing that \(e^x\) must equal \(e^3\). The natural log is powerful in converting exponential relationships into linear ones, making them easier to solve.

Simplifying Equations

Simplifying equations involves breaking them down into the simplest form without changing their value. This is often a necessary step when solving both exponential and logarithmic equations, as it allows for easier manipulation and understanding.

• In our equation, \(\ln e^x = 3\), simplification leads us to \(e^x = e^3\).
• Recognizing that if the bases are the same (\(e\)), the exponents must be equal, so \(x = 3\).

Always look for opportunities to apply properties of exponents and logarithms to simplify. Simplification can convert what appears to be a complex equation into an easily solved format, thus streamlining the problem-solving process. The ability to simplify efficiently is a skill that will be beneficial in all areas of mathematics.