Problem 57
Question
In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 3 \ln x-\frac{1}{3} \ln y $$
Step-by-Step Solution
Verified Answer
The simplified expression is \(\ln(\frac{x^3}{y^{\frac{1}{3}}})\)
1Step 1: Apply the rule of exponents to both terms
The expression can be rewritten as \(\ln(x^3) - \ln(y^{1/3})\)
2Step 2: Apply the subtraction rule
The expression can be further simplified to \(\ln(\frac{x^3}{y^{1/3}})\)
Key Concepts
Properties of LogarithmsCondensing Logarithmic ExpressionsExponential RulesSubtraction Rule in Logarithms
Properties of Logarithms
Logarithms have several useful properties that make them quite powerful in mathematical expressions. Two of the essential properties are the product rule and the power rule.
The product rule tells us that the logarithm of a product is the sum of the logarithms:
The power rule helps us manage exponents conveniently:
These properties allow us to simplify and manipulate logarithmic expressions, making our calculations more manageable.
The product rule tells us that the logarithm of a product is the sum of the logarithms:
- \( \log_b(mn) = \log_b(m) + \log_b(n) \)
The power rule helps us manage exponents conveniently:
- \( \log_b(m^n) = n \cdot \log_b(m) \)
These properties allow us to simplify and manipulate logarithmic expressions, making our calculations more manageable.
Condensing Logarithmic Expressions
Condensing logarithms involves combining multiple logarithmic terms into a single term. This process uses the properties of logarithms to rewrite expressions compactly.
For example, you might encounter expressions like \( \ln a + \ln b \) and condense them to \( \ln(ab) \). The goal is to transform expressions using the rules systematically, resulting in a more simplified form.
The condensing process is essential in solving equations where working with a single logarithmic term is easier and more efficient.
For example, you might encounter expressions like \( \ln a + \ln b \) and condense them to \( \ln(ab) \). The goal is to transform expressions using the rules systematically, resulting in a more simplified form.
The condensing process is essential in solving equations where working with a single logarithmic term is easier and more efficient.
Exponential Rules
The process of converting expressions involves understanding the rules of exponents, especially when working with logarithms.
In expressions like \( 3 \ln x \), you can rewrite them using the power rule:
Similarly, \( \frac{1}{3} \ln y \) can be rewritten as \( \ln(y^{1/3}) \).
These transformations help in rewriting log expressions as one single term, simplifying further operations such as subtraction, as we've seen in this exercise.
In expressions like \( 3 \ln x \), you can rewrite them using the power rule:
- \( \ln(x^3) \)
Similarly, \( \frac{1}{3} \ln y \) can be rewritten as \( \ln(y^{1/3}) \).
These transformations help in rewriting log expressions as one single term, simplifying further operations such as subtraction, as we've seen in this exercise.
Subtraction Rule in Logarithms
The subtraction rule in logarithms helps us understand the difference between two logarithmic expressions. This rule says:
Using this rule, if you have an expression like \( \ln(x^3) - \ln(y^{1/3}) \), it becomes \( \ln\left(\frac{x^3}{y^{1/3}}\right) \), successfully condensing the terms.
This rule is handy for simplifying expressions and solving equations that involve differences between logs, making it an essential aspect of logarithmic calculations.
- \( \log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right) \)
Using this rule, if you have an expression like \( \ln(x^3) - \ln(y^{1/3}) \), it becomes \( \ln\left(\frac{x^3}{y^{1/3}}\right) \), successfully condensing the terms.
This rule is handy for simplifying expressions and solving equations that involve differences between logs, making it an essential aspect of logarithmic calculations.
Other exercises in this chapter
Problem 56
In \(1626,\) Peter Minuit convinced the Wappinger Indians to sell him Manhattan Island for \(\$ 24 .\) If the Native Americans had put the \(\$ 24\) into a bank
View solution Problem 56
Find the domain of each logarithmic function. $$f(x)=\log _{5}(x+6)$$
View solution Problem 57
The function $$N(t)=\frac{30,000}{1+20 e^{-1.5 t}}$$ describes the number of people, \(N(t),\) who become ill with influenza \(t\) weeks after its initial outbr
View solution Problem 57
Find the domain of each logarithmic function. $$f(x)=\log (2-x)$$
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