Problem 27
Question
In Exercises, apply the inverse properties of logarithmic and exponential functions to simplify the expression. $$ -1+\ln e^{2 x} $$
Step-by-Step Solution
Verified Answer
The simplified form of the expression \( -1+\ln e^{2 x} \) is \( -1 + 2x \).
1Step 1: Identify the logarithmic and exponential functions
The term \( \ln e^{2x} \) in the expression involves both a natural logarithm function and an exponential function. The natural logarithm \( \ln \) is the inverse of the exponential \( e^x \). Therefore, applying the inverse property here would simplify this term.
2Step 2: Apply the exponential and logarithmic inverse property
The inverse property states that \( \ln e^x = x \). Therefore, \( \ln e^{2x} = 2x \). So, the original expression simplifies to \( -1 + 2x \).
3Step 3: Final Simplified Expression
The final simplified expression is therefore \( -1 + 2x \).
Key Concepts
Natural LogarithmExponential FunctionSimplification of Expressions
Natural Logarithm
The natural logarithm, represented by \( \ln \), is a mathematical operation that helps us find the power to which the base \( e \) must be raised to obtain a given number. This base \( e \) is an important constant, approximately equal to 2.718, commonly used in mathematical modeling. The natural logarithm is particularly popular due to its simplicity when related to exponential growth and decay models.
To understand how it simplifies expressions, think of the inverse relationship it has with the exponential function. Essentially, \( \ln(a) \) answers the question: "To what power should \( e \) be raised, to yield \( a \)?"
The natural logarithm has several key properties, such as:
To understand how it simplifies expressions, think of the inverse relationship it has with the exponential function. Essentially, \( \ln(a) \) answers the question: "To what power should \( e \) be raised, to yield \( a \)?"
The natural logarithm has several key properties, such as:
- \( \ln(1) = 0 \) because \( e^0 = 1 \)
- \( \ln(e) = 1 \) because \( e^1 = e \)
- \( \ln(e^x) = x \) due to the inverse nature with \( e^x \)
Exponential Function
An exponential function is a type of mathematical function in the form of \( f(x) = a \cdot e^{bx} \), where \( e \) is the base of natural logarithms and \( a \) and \( b \) are constants. Exponential functions model dynamic systems that grow or decay at rates proportional to their current value, like population growth or radioactive decay.
The magic of the exponential function lies in its predictable pattern of growth. With the special base \( e \), this function becomes its own derivative, which is invaluable in calculus and differential equations.
The main utility of exponential functions in simplification problems is through their inverse relationship with logarithms. This relationship allows us to "unravel" complex expressions with ease. For example, the equation \( e^{2x} \) can be simplified using the natural logarithm to solve for the exponent directly by using the property that \( \ln(e^{2x}) = 2x \).
It's this property that makes exponential functions and natural logarithms such powerful tools in algebra and beyond. By recognizing this relationship, simplification becomes a straightforward process.
The magic of the exponential function lies in its predictable pattern of growth. With the special base \( e \), this function becomes its own derivative, which is invaluable in calculus and differential equations.
The main utility of exponential functions in simplification problems is through their inverse relationship with logarithms. This relationship allows us to "unravel" complex expressions with ease. For example, the equation \( e^{2x} \) can be simplified using the natural logarithm to solve for the exponent directly by using the property that \( \ln(e^{2x}) = 2x \).
It's this property that makes exponential functions and natural logarithms such powerful tools in algebra and beyond. By recognizing this relationship, simplification becomes a straightforward process.
Simplification of Expressions
Simplifying expressions is a central task in algebra that involves rewriting them in their simplest form. This process typically makes mathematical operations easier and more efficient by reducing complex terms into basic components.
When dealing with logarithmic and exponential expressions, simplification often utilizes their inverse properties. For instance, knowing that \( \ln(e^x) = x \) allows us to remove complicated layers within expressions by applying these properties.
In the given exercise, the expression \( -1 + \ln e^{2x} \) was simplified using these principles. The term \( \ln e^{2x} \) was directly resolved to \( 2x \) utilizing the inverse property, which states that the logarithm of a number raised to a power returns the power itself.
Simplification of expressions isn't simply about making them look smaller; it's about enhancing their usability. This reduction helps in solving equations, plotting graphs, or even evaluating functions with ease. By stripping down expressions to their simplest form, we are better equipped to understand, analyze, and communicate mathematical ideas.
When dealing with logarithmic and exponential expressions, simplification often utilizes their inverse properties. For instance, knowing that \( \ln(e^x) = x \) allows us to remove complicated layers within expressions by applying these properties.
In the given exercise, the expression \( -1 + \ln e^{2x} \) was simplified using these principles. The term \( \ln e^{2x} \) was directly resolved to \( 2x \) utilizing the inverse property, which states that the logarithm of a number raised to a power returns the power itself.
Simplification of expressions isn't simply about making them look smaller; it's about enhancing their usability. This reduction helps in solving equations, plotting graphs, or even evaluating functions with ease. By stripping down expressions to their simplest form, we are better equipped to understand, analyze, and communicate mathematical ideas.
Other exercises in this chapter
Problem 26
In Exercises, find \(d y / d x\) implicitly. $$ e^{x y}+x^{2}-y^{2}=10 $$
View solution Problem 26
In Exercises, use a graphing utility to graph the function. $$ y=-5^{x} $$
View solution Problem 27
In Exercises, find the second derivative. $$ f(x)=2 e^{3 x}+3 e^{-2 x} $$
View solution Problem 27
In Exercises, use a graphing utility to graph the function. $$ y=3^{-x^{2}} $$
View solution