Problem 37
Question
Given \(\ln 4=1.3863\) and \(\ln 5=1.6094,\) find each value. Do not use a calculator. $$ \ln \sqrt{e^{8}} $$
Step-by-Step Solution
Verified Answer
The value is 4.
1Step 1: Simplify the Inner Expression
The expression inside the natural logarithm is \( \sqrt{e^8} \). We know that \( \sqrt{x} = x^{1/2} \), so we can rewrite \( \sqrt{e^8} \) as \( (e^8)^{1/2} \). Using the property of exponents, this becomes \( e^{8/2} = e^4 \).
2Step 2: Apply the Logarithm Property
Now, we have \( \ln(e^4) \). Using the logarithm power rule, \( \ln(b^x) = x \cdot \ln(b) \), we can rewrite this as \( 4 \cdot \ln(e) \). Since \( \ln(e) = 1 \), this simplifies to \( 4 \cdot 1 \).
3Step 3: Final Computation
Substituting the result from the previous step, we have \( 4 \cdot 1 = 4 \). So, \( \ln \sqrt{e^8} = 4 \).
Key Concepts
ExponentsLogarithm PropertiesSimplification
Exponents
Exponents are a way to express repeated multiplication of the same number. For example, the expression \(e^8\) means you multiply the number \(e\), which is approximately 2.718, by itself 8 times. Exponents are crucial in mathematics as they simplify repeated multiplications.
When you encounter a square root like \(\sqrt{e^8}\), you can use the property that a square root is equivalent to raising a number to the power of \(1/2\). This gives us \(\sqrt{e^8} = (e^8)^{1/2}\). The law of exponents then tells us that when raising a power to another power, you multiply the exponents, yielding \(e^{8 \cdot 1/2} = e^4\). This transformation helps greatly in dealing with complex expressions like those involving natural logarithms.
When you encounter a square root like \(\sqrt{e^8}\), you can use the property that a square root is equivalent to raising a number to the power of \(1/2\). This gives us \(\sqrt{e^8} = (e^8)^{1/2}\). The law of exponents then tells us that when raising a power to another power, you multiply the exponents, yielding \(e^{8 \cdot 1/2} = e^4\). This transformation helps greatly in dealing with complex expressions like those involving natural logarithms.
Logarithm Properties
Logarithms are the inverse operation of exponents. The natural logarithm, \(\ln\), specifically uses the base \(e\). One crucial property of logarithms is how they handle exponents. The power rule of logarithms states that \(\ln(b^x) = x \cdot \ln(b)\). This property allows us to simplify expressions that seem complex at first glance.
In the given problem, after simplifying the inner expression to \(e^4\), we use the power rule to transform \(\ln(e^4)\) into \(4 \cdot \ln(e)\). Since we know \(\ln(e) = 1\) (because the natural logarithm of its base, \(e\), is one), the computation becomes straightforward. The properties of logarithms make it easier to work with elaborate exponential expressions, allowing us to convert them into simpler calculations.
In the given problem, after simplifying the inner expression to \(e^4\), we use the power rule to transform \(\ln(e^4)\) into \(4 \cdot \ln(e)\). Since we know \(\ln(e) = 1\) (because the natural logarithm of its base, \(e\), is one), the computation becomes straightforward. The properties of logarithms make it easier to work with elaborate exponential expressions, allowing us to convert them into simpler calculations.
Simplification
Simplification is about reducing expressions to their most basic form to make calculations easier. In this exercise, simplification involves three main steps.
The goal of simplification is to perform calculations in a way that is more intuitive and less prone to error. By breaking down problems into simpler components and using well-defined rules, we can achieve neat solutions without relying on calculators. This not only helps in exams but also strengthens conceptual understanding.
- First, transforming \(\sqrt{e^8}\) to \(e^4\).
- Second, applying the power rule of logarithms to express \(\ln(e^4)\) as \(4 \cdot \ln(e)\).
- Finally, substituting the value of \(\ln(e) = 1\), which turns \(4 \cdot 1\) into just 4.
The goal of simplification is to perform calculations in a way that is more intuitive and less prone to error. By breaking down problems into simpler components and using well-defined rules, we can achieve neat solutions without relying on calculators. This not only helps in exams but also strengthens conceptual understanding.
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