Problem 90
Question
In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \ln e^{7} $$
Step-by-Step Solution
Verified Answer
The simplification of \( \ln e^{7} \) is 7.
1Step 1: Identify the expressions
Given an expression \( \ln e^{7} \). It is in the form of \( \ln(a^b) \). Here, \( a = e \), \( b = 7 \).
2Step 2: Apply Property of Logarithms
Now, using the logarithm property \( \ln(a^b) = b \ln a \), and substituting values we get: \( 7 \ln e \).
3Step 3: Applying the Equation of Natural Logarithms
The logarithm base \( e \) of \( e \) is equals to 1, hence, \(\ln e = 1\). Substituting into above, we get \( 7 \times 1 \).
4Step 4: Simplification
Finally, we simplify the expression \( 7 \times 1 = 7 \).
Key Concepts
Properties of LogarithmsNatural Logarithm PropertiesLogarithm SimplificationExponential and Logarithmic Expressions
Properties of Logarithms
Understanding the properties of logarithms is crucial for simplifying logarithmic expressions effectively. In essence, logarithms tell us the power to which a number (the base) must be raised to produce another number. Logarithms have several unique properties that can make complex problems much simpler.
One essential property is the power rule, represented as \( \log_b(a^n) = n \log_b(a) \). This property states that a logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. Another property is the product rule, \( \log_b(xy) = \log_b(x) + \log_b(y) \), asserting that the logarithm of a product is the sum of the logarithms.
When working with logarithms, also consider the quotient rule, \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \), and the base-change rule, which provides a method for converting logs with one base to logs with another. These properties are foundational and universally applicable across different logarithm bases, including the natural logarithm, denoted as \( \ln \).
One essential property is the power rule, represented as \( \log_b(a^n) = n \log_b(a) \). This property states that a logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. Another property is the product rule, \( \log_b(xy) = \log_b(x) + \log_b(y) \), asserting that the logarithm of a product is the sum of the logarithms.
When working with logarithms, also consider the quotient rule, \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \), and the base-change rule, which provides a method for converting logs with one base to logs with another. These properties are foundational and universally applicable across different logarithm bases, including the natural logarithm, denoted as \( \ln \).
Natural Logarithm Properties
The natural logarithm, \( \ln \), is a special instance of logarithms where the base is the mathematical constant \( e \), approximately equal to 2.71828. The properties of \( \ln \), mirror the properties of general logarithms but relate specifically to the base \( e \).
One unique natural logarithm property is \( \ln(e) = 1 \), which is crucial in simplification processes. The power rule for natural logarithms becomes \( \ln(e^n) = n \), indicating the natural logarithm of \( e \), raised to any power, is simply that power. Conversely, the expression \( e^{\ln(x)} = x \) shows the inverse relationship between the exponential function and the natural logarithm.
These rules are not just academic; they underpin numerous calculations in fields like calculus and complex number analysis where \( e \), and \( \ln \) routinely appear. For students, mastering and applying these properties can substantially reduce the complexity of many homework problems.
One unique natural logarithm property is \( \ln(e) = 1 \), which is crucial in simplification processes. The power rule for natural logarithms becomes \( \ln(e^n) = n \), indicating the natural logarithm of \( e \), raised to any power, is simply that power. Conversely, the expression \( e^{\ln(x)} = x \) shows the inverse relationship between the exponential function and the natural logarithm.
These rules are not just academic; they underpin numerous calculations in fields like calculus and complex number analysis where \( e \), and \( \ln \) routinely appear. For students, mastering and applying these properties can substantially reduce the complexity of many homework problems.
Logarithm Simplification
Logarithm simplification involves using the properties of logarithms to condense or break down expressions into more manageable forms. This process is not only key to solving logarithmic equations but also enhances our ability to understand the relationship between logarithmic and exponential forms.
When simplifying expressions like \( \ln e^{7} \), we systematically apply the natural logarithm properties. Our first step is typically to look for exponents that can be moved out front, this transformation directly relies on the power rule of logarithms. Then, we consider if the base of the logarithm and the number it's operating on share a common base—if so, it simplifies nicely, as we've seen with expressions involving \( e \).
When simplifying expressions like \( \ln e^{7} \), we systematically apply the natural logarithm properties. Our first step is typically to look for exponents that can be moved out front, this transformation directly relies on the power rule of logarithms. Then, we consider if the base of the logarithm and the number it's operating on share a common base—if so, it simplifies nicely, as we've seen with expressions involving \( e \).
Exponential and Logarithmic Expressions
Exponential and logarithmic expressions are deeply intertwined, with each representing the inverse of the other. Understanding the connection between these two types of expressions is vital to mastering calculus and algebra.
The expression \( e^{x} \) is the exponential form where \( e \) is the base and \( x \) is the exponent. The inverse of this, \( \ln(x) \), asks the question: 'To what power must we raise \( e \) to obtain \( x \)?' It's a fundamental concept that plays out in natural growth and decay models, among others.
Transforming between exponential and logarithmic forms is essentially about changing perspectives. When you encounter an expression like \( e^{\ln(x)} \), you're essentially looking at a circular operation, where the \( e \) and \( \ln \) cancel out, reflecting their inverse relationship and leaving you simply with \( x \). This concept underlies the simplification process, giving structure to what might otherwise seem like an abstract mathematical exercise.
The expression \( e^{x} \) is the exponential form where \( e \) is the base and \( x \) is the exponent. The inverse of this, \( \ln(x) \), asks the question: 'To what power must we raise \( e \) to obtain \( x \)?' It's a fundamental concept that plays out in natural growth and decay models, among others.
Transforming between exponential and logarithmic forms is essentially about changing perspectives. When you encounter an expression like \( e^{\ln(x)} \), you're essentially looking at a circular operation, where the \( e \) and \( \ln \) cancel out, reflecting their inverse relationship and leaving you simply with \( x \). This concept underlies the simplification process, giving structure to what might otherwise seem like an abstract mathematical exercise.
Other exercises in this chapter
Problem 89
In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \ln e^{20} $$
View solution Problem 90
Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer.
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determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The functions \(f(x)=\le
View solution Problem 91
Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s)
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