Problem 43
Question
Use properties of logarithms to condense logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. \(\ln x+\ln 7\)
Step-by-Step Solution
Verified Answer
The simplified single logarithm of the expression \(\ln x + \ln 7\) is \(\ln 7x\).
1Step 1: Identify the form of the given problem
The problem here is a logarithmic expression of the form \(\ln m + \ln n\) which can be condensed using the logarithm properties.
2Step 2: Apply the appropriate logarithm property
We apply the property of logarithms that states \(\ln m + \ln n = \ln (mn)\). So \(\ln x + \ln 7 = \ln (x*7)\).
3Step 3: Simplify the expression
Simplify \(x * 7\) to \(7x\). So the expression \(\ln x + \ln 7\) simplifies to \(\ln 7x\).
Key Concepts
Properties of LogarithmsNatural LogarithmLogarithmic IdentitiesSimplifying Logarithms
Properties of Logarithms
Understanding the properties of logarithms is essential when dealing with complex logarithmic expressions. One of the fundamental properties is the product rule, which tells us how to condense two logs with the same base when they are added together. The rule states that \( \log_b(m) + \log_b(n) = \log_b(mn) \), where \(b\) is the base, and \(m\) and \(n\) are the numbers we're taking the log of.
Another important property is the quotient rule. It helps to simplify the difference of two logarithms with the same base: \( \log_b(m) - \log_b(n) = \log_b(\frac{m}{n}) \). Finally, the power rule simplifies the multiplication of a logarithm by a constant, stating that \( c \cdot \log_b(m) = \log_b(m^c) \).
Using these properties, we can combine and simplify logarithmic expressions, turning several terms into a single term with a base of 1, which makes calculations more manageable, especially when we can't use calculators.
Another important property is the quotient rule. It helps to simplify the difference of two logarithms with the same base: \( \log_b(m) - \log_b(n) = \log_b(\frac{m}{n}) \). Finally, the power rule simplifies the multiplication of a logarithm by a constant, stating that \( c \cdot \log_b(m) = \log_b(m^c) \).
Using these properties, we can combine and simplify logarithmic expressions, turning several terms into a single term with a base of 1, which makes calculations more manageable, especially when we can't use calculators.
Natural Logarithm
The natural logarithm, represented by \( \ln \), uses the constant \(e\) as its base and is widely used in mathematics, especially in calculus and physics. The constant \(e\) is approximately 2.71828, and it has unique growth properties that make it naturally occurring in various scientific laws and growth patterns.
The natural logarithm has properties similar to logarithms with other bases, but it simplifies expressions involving the number \(e\) because \( \ln(e^x) = x \) by definition. Therefore, \( \ln \) is a profound tool in solving equations and analyzing change because its base \(e\) is an integral part of exponential growth and decay models.
The natural logarithm has properties similar to logarithms with other bases, but it simplifies expressions involving the number \(e\) because \( \ln(e^x) = x \) by definition. Therefore, \( \ln \) is a profound tool in solving equations and analyzing change because its base \(e\) is an integral part of exponential growth and decay models.
Logarithmic Identities
Several logarithmic identities are universally accepted and incredibly useful for manipulating and simplifying logarithmic expressions. Important identities include:
These identities serve as shortcuts and allow us to simplify expressions or solve logarithmic equations without lengthy calculations. They are essential in understanding deeper log properties and are often applied in conjunction with the rules of logarithms to tackle more complex problems.
- \(\log_b(b) = 1\)
- \(\log_b(1) = 0\)
- \(\log_b(b^x) = x\)
- For any positive number \(a\), \(\log_b(a^x) = x \cdot \log_b(a)\)
These identities serve as shortcuts and allow us to simplify expressions or solve logarithmic equations without lengthy calculations. They are essential in understanding deeper log properties and are often applied in conjunction with the rules of logarithms to tackle more complex problems.
Simplifying Logarithms
Simplifying logarithms involves reducing complex logarithmic expressions into simpler forms using the properties and identities of logarithms. Exercises like the provided example \(\ln x + \ln 7\) are straightforward once you understand the relevant rules.
Following the product rule for simplification, as shown in the step-by-step solution, condenses the expression from two separate natural logs into one. From a teaching perspective, it's important to encourage students to look for such patterns where logarithmic properties can apply. This skill not only simplifies their work but also helps them develop a deeper understanding of the underlying mathematical concepts.
Always remind students to be cautious with the applicable range of the properties, such as ensuring the logs have the same base and the values inside the logs are positive, as these are common points where mistakes can occur.
Following the product rule for simplification, as shown in the step-by-step solution, condenses the expression from two separate natural logs into one. From a teaching perspective, it's important to encourage students to look for such patterns where logarithmic properties can apply. This skill not only simplifies their work but also helps them develop a deeper understanding of the underlying mathematical concepts.
Always remind students to be cautious with the applicable range of the properties, such as ensuring the logs have the same base and the values inside the logs are positive, as these are common points where mistakes can occur.
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