Problem 54
Question
Simplify the expression. $$3-\ln \left(e^{x^{2}+2}\right)$$
Step-by-Step Solution
Verified Answer
The simplified expression is \(1-x^2\).
1Step 1: Simplify Expression
The first step is to recognize that the natural logarithm and the exponential function with base 'e' are inverse functions. When they are paired, they cancel each other out. Applying this property to the given expression, it simplifies to: \(3-\ln \left(e^{x^{2}+2}\right)\) simplifies to \(3-(x^2+2)\).
2Step 2: Simplify the Remaining Algebraic Expression
After applying the property of logarithms, the expression turns into an algebraic expression which when simplified gives: \(3-(x^2+2)\) simplifies to \(1-x^2\).
Key Concepts
Exponential FunctionsInverse FunctionsNatural LogarithmAlgebraic Expressions
Exponential Functions
Exponential functions have the form \( f(x) = a^x \), where \( a \) is a positive constant. The most common base for \( a \) is the mathematical constant 'e', approximately equal to 2.71828. This special base is used in natural exponential functions, denoted as \( e^x \). These functions are essential in many natural phenomena such as population growth and compound interest.
Exponential functions are defined so that any power of \( e \), like \( e^{x^2 + 2} \), grows rapidly as the value of the exponent increases. They are continuous and smooth, without any breaks or sharp turns. Understanding their behavior is crucial when dealing with problems involving exponential growth or decay.
Remember, exponential functions grow exponentially: a small increase in \( x \) results in a much larger increase in \( e^x \). This is why they're widely applicable in real-world scenarios.
Exponential functions are defined so that any power of \( e \), like \( e^{x^2 + 2} \), grows rapidly as the value of the exponent increases. They are continuous and smooth, without any breaks or sharp turns. Understanding their behavior is crucial when dealing with problems involving exponential growth or decay.
Remember, exponential functions grow exponentially: a small increase in \( x \) results in a much larger increase in \( e^x \). This is why they're widely applicable in real-world scenarios.
Inverse Functions
Inverse functions reverse the process of their original functions. If a function \( f \) converts an input \( x \) into an output \( y \), its inverse \( f^{-1} \) will convert \( y \) back into \( x \). An important property of inverse functions is that they "undo" each other.
The inverse function of an exponential function \( e^x \) is the natural logarithm \( \, \ln x \). This means that if you apply \( \, \ln \) to \( e^x \), you get \( x \); mathematically, \( \, \ln(e^x) = x \).
In the exercise, the natural logarithm function \( \, \ln \) and the exponential function \( e^{x^2 + 2} \) are inverses. Therefore, they cancel each other out: \( \, \ln(e^{x^2 + 2}) = x^2 + 2 \). This simplification step is crucial when solving complex expressions.
The inverse function of an exponential function \( e^x \) is the natural logarithm \( \, \ln x \). This means that if you apply \( \, \ln \) to \( e^x \), you get \( x \); mathematically, \( \, \ln(e^x) = x \).
In the exercise, the natural logarithm function \( \, \ln \) and the exponential function \( e^{x^2 + 2} \) are inverses. Therefore, they cancel each other out: \( \, \ln(e^{x^2 + 2}) = x^2 + 2 \). This simplification step is crucial when solving complex expressions.
Natural Logarithm
The natural logarithm, denoted as \( \, \ln(x) \), is one of the most useful logarithm functions in mathematics. It is the inverse operation of taking the power of \( e \). In simpler terms, it tells you how many times you need to multiply \( e \) to get a particular number.
The key characteristic of the natural logarithm is its ability to transform multiplicative processes into additive ones. For example, \( \, \ln(a \, \cdot b) = \, \ln(a) + \, \ln(b) \). This property makes it easier to solve complex multiplication, especially when dealing with exponential growth.
In mathematical problems like our exercise, the natural logarithm is paired with exponentials to simplify terms. Its main role is to manage and unravel exponential expressions, making them more manageable.
The key characteristic of the natural logarithm is its ability to transform multiplicative processes into additive ones. For example, \( \, \ln(a \, \cdot b) = \, \ln(a) + \, \ln(b) \). This property makes it easier to solve complex multiplication, especially when dealing with exponential growth.
In mathematical problems like our exercise, the natural logarithm is paired with exponentials to simplify terms. Its main role is to manage and unravel exponential expressions, making them more manageable.
Algebraic Expressions
Algebraic expressions involve numbers, variables, and the operations of addition, subtraction, multiplication, and division arranged into a meaningful expression. Simplifying these expressions is a skill that helps in solving equations efficiently.
In the exercise early on, after using the inverse property of the \( \, \ln \) and \( e^x \), the remaining expression \( 3 - (x^2 + 2) \) becomes purely algebraic. Simplifying it requires applying basic algebraic principles by distributing the negative sign: \( 3 - x^2 - 2 \).
After simplifying further, you arrive at \( 1 - x^2 \). This outcome underlines the importance of step-by-step simplification, showing that complex-looking expressions can often reduce to simpler forms. This skill is vital when solving mathematical problems, making sure the final answer is as clear as possible.
In the exercise early on, after using the inverse property of the \( \, \ln \) and \( e^x \), the remaining expression \( 3 - (x^2 + 2) \) becomes purely algebraic. Simplifying it requires applying basic algebraic principles by distributing the negative sign: \( 3 - x^2 - 2 \).
After simplifying further, you arrive at \( 1 - x^2 \). This outcome underlines the importance of step-by-step simplification, showing that complex-looking expressions can often reduce to simpler forms. This skill is vital when solving mathematical problems, making sure the final answer is as clear as possible.
Other exercises in this chapter
Problem 54
Use the acidity model \(\mathbf{p H}=-\log \left[\mathbf{H}^{+}\right]\) where acidity (pH) is a measure of the hydrogen ion concentration \(\left[\mathbf{H}^{+
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Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\
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Use a graphing utility to (a) graph the function and (b) find any asymptotes numerically by creating a table of values for the function. $$g(x)=\frac{8}{1+e^{-0
View solution Problem 55
Use the acidity model \(\mathbf{p H}=-\log \left[\mathbf{H}^{+}\right]\) where acidity (pH) is a measure of the hydrogen ion concentration \(\left[\mathbf{H}^{+
View solution