Problem 24
Question
In Exercises, apply the inverse properties of logarithmic and exponential functions to simplify the expression. $$ \ln e^{2 x-1} $$
Step-by-Step Solution
Verified Answer
The simplified expression is \(2x-1\).
1Step 1: Apply the relevant algebraic rules
Use properties of exponents, radicals, or algebraic identities to simplify the expression.
2Step 2: State the simplified result
The simplified expression is The simplified expression is \(2x-1\)..
Key Concepts
Logarithmic FunctionsExponential FunctionsNatural Logarithm
Logarithmic Functions
Logarithmic functions help us understand exponential relationships in mathematics. They are the opposite, or inverse, of exponential functions. If you have an exponential function expressed as \(y = a^x\), then the corresponding logarithmic function would be \(x = \log_a(y)\). In simple terms, logarithms tell us the power to which we need to raise a base number to get another number.
The inverse nature of logarithms means that they can "undo" the exponentiation. For instance, in the expression \(\ln(e^{2x-1})\), the \(\ln\) (natural logarithm) and \(e\) (Euler's number) cancel each other out, simplifying the expression to \(2x-1\). This is a crucial property, as it can simplify complex expressions easily, making calculations more manageable.
The inverse nature of logarithms means that they can "undo" the exponentiation. For instance, in the expression \(\ln(e^{2x-1})\), the \(\ln\) (natural logarithm) and \(e\) (Euler's number) cancel each other out, simplifying the expression to \(2x-1\). This is a crucial property, as it can simplify complex expressions easily, making calculations more manageable.
Exponential Functions
Exponential functions are mathematical expressions that have a constant base raised to a variable exponent. These functions are key to understanding phenomena that grow or decay at a steady rate, such as population growth or radioactive decay. A general form of an exponential function is \(f(x) = a^x\), where \(a\) is a positive constant.
These functions are "one-to-one," which means each input \(x\) gives a unique output \(f(x)\). They are especially important because they have a direct inverse - logarithmic functions. For example, the function \(e^x\) has an inverse \(\ln x\). In expressions like \(e^{2x-1}\), when paired with its inverse logarithmic function, it simplifies to the exponent, showing how powerful these inverses are for solving equations and simplifying math problems.
These functions are "one-to-one," which means each input \(x\) gives a unique output \(f(x)\). They are especially important because they have a direct inverse - logarithmic functions. For example, the function \(e^x\) has an inverse \(\ln x\). In expressions like \(e^{2x-1}\), when paired with its inverse logarithmic function, it simplifies to the exponent, showing how powerful these inverses are for solving equations and simplifying math problems.
Natural Logarithm
The natural logarithm, denoted as \(\ln\), is a specific logarithm that uses Euler's number \(e\) (approximately 2.71828) as the base. It is widely used in various fields, from economics to scientific calculations, due to its natural occurrence in growth processes.
Using the natural logarithm allows us to simplify complex exponential expressions. For instance, in the expression \(\ln(e^{2x-1})\), the natural logarithm's property as the inverse of exponential functions becomes evident. When you apply \(\ln\) to \(e^{2x-1}\), it cancels out the \(e\), and you're left with the exponent \(2x-1\). This powerful property transforms difficult calculations into simpler algebraic expressions, facilitating easier understanding and manipulation of exponential growth or decay problems.
Using the natural logarithm allows us to simplify complex exponential expressions. For instance, in the expression \(\ln(e^{2x-1})\), the natural logarithm's property as the inverse of exponential functions becomes evident. When you apply \(\ln\) to \(e^{2x-1}\), it cancels out the \(e\), and you're left with the exponent \(2x-1\). This powerful property transforms difficult calculations into simpler algebraic expressions, facilitating easier understanding and manipulation of exponential growth or decay problems.
Other exercises in this chapter
Problem 23
In Exercises, use a graphing utility to graph the function. $$ y=2^{x-1} $$
View solution Problem 24
In Exercises, find the derivative of the function. $$ f(x)=x \ln e^{x^{2}} $$
View solution Problem 24
In Exercises, find \(d y / d x\) implicitly. $$ x^{2} y-e^{y}-4=0 $$
View solution Problem 24
Use a graphing utility to graph the function. Describe the shape of the graph for very large and very small values of \(x\). (a) \(f(x)=\frac{8}{1+e^{-0.5 x}}\)
View solution