Problem 93
Question
Which expression is equal to \(\log _{5} x+4 \cdot \log _{5} y-2 \cdot \log _{5} z ?\) \(\begin{array}{llll}{\text { A. } \log _{5}(-8 x y z)} & {\text { B. }-\log _{5} \frac{4 x y}{2 z}} & {\text { C. } \log _{5} \frac{(x y)^{4}}{z^{2}}} & {\text { D. } \log _{5} \frac{x y^{4}}{z^{2}}}\end{array}\)
Step-by-Step Solution
Verified Answer
The correct answer is option D: \( \log _{5} \frac{x y^{4}}{z^{2}} \).
1Step 1: Apply the Power Rule for Logarithms
The given expression can be simplified using the properties of logarithms, particularly the power rule. The power rule states that for any logarithm, including \( \log _{5} x \), the logarithm of a number raised to a power is equal to the power times the logarithm of the number itself. Hence, the given expression can be expressed as: \( \log _{5} x+ \log _{5} y^{4} - \log _{5} z^{2} \).
2Step 2: Apply the Division Rule for Logarithms
Following the power rule simplification, we now have a sum and a subtraction of logarithms. We can simplify this further by using the division rule, which states that the logarithm of a division of two numbers is equal to the logarithm of the numerator minus the logarithm of the denominator. This leads us to: \( \log _{5} \frac{x y^{4}}{z^{2}} \).
3Step 3: Match the Simplified Expression to the Provided Options
Upon comparing our simples expression to the provided answer choices, it is clear that it matches with answer option D. Therefore, option D: \( \log _{5} \frac{x y^{4}}{z^{2}} \) is the correct answer to this problem. The other options cannot be correct as they do not match the simplified expression.
Key Concepts
Power Rule for LogarithmsDivision Rule for LogarithmsProperties of LogarithmsSimplifying Logarithmic Expressions
Power Rule for Logarithms
The power rule for logarithms is a fundamental property that simplifies expressions where a logarithm has a number raised to a power. This rule states that the logarithm of a power can be rewritten as the exponent multiplied by the logarithm itself: \( \log_b(x^n) = n \cdot \log_b(x) \).
For example, if you have \( \log_5(y^4) \), you can use the power rule to express it as \( 4 \cdot \log_5(y) \). This transformation helps in handling complex logarithmic expressions by deconstructing them into simpler parts.
In our exercise, the expression \( 4 \cdot \log_5(y) \) comes from applying this rule to \( \log_5(y^4) \), which makes it easier to work with other rules and continue simplifying.
For example, if you have \( \log_5(y^4) \), you can use the power rule to express it as \( 4 \cdot \log_5(y) \). This transformation helps in handling complex logarithmic expressions by deconstructing them into simpler parts.
In our exercise, the expression \( 4 \cdot \log_5(y) \) comes from applying this rule to \( \log_5(y^4) \), which makes it easier to work with other rules and continue simplifying.
Division Rule for Logarithms
The division rule for logarithms focuses on the relationship between the logarithm of a division and subtraction of two logarithms. The rule is expressed as follows: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \).
Using this property helps to reduce expressions especially when they involve both numerators and denominators in problem-solving situations.
In our example, after applying the power rule, the resulting expression \( \log_5(x) + \log_5(y^4) - \log_5(z^2) \) can be transformed using the division rule to \( \log_5\left(\frac{x \cdot y^4}{z^2}\right) \). This conversion is crucial for comparing the simplified logarithmic expression with multiple choice options.
Using this property helps to reduce expressions especially when they involve both numerators and denominators in problem-solving situations.
In our example, after applying the power rule, the resulting expression \( \log_5(x) + \log_5(y^4) - \log_5(z^2) \) can be transformed using the division rule to \( \log_5\left(\frac{x \cdot y^4}{z^2}\right) \). This conversion is crucial for comparing the simplified logarithmic expression with multiple choice options.
Properties of Logarithms
Logarithms have several important properties that allow for effective simplification and manipulation of expressions. These include:
Rich in versatility, these principles enable handling of logarithms in a similar way to how other algebraic expressions are managed, making them valuable tools in both academic and real-world applications.
- Product Property: \( \log_b(x) + \log_b(y) = \log_b(xy) \)
- Quotient Property: \( \log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right) \)
- Power Property: \( \log_b(x^n) = n \cdot \log_b(x) \)
Rich in versatility, these principles enable handling of logarithms in a similar way to how other algebraic expressions are managed, making them valuable tools in both academic and real-world applications.
Simplifying Logarithmic Expressions
The process of simplifying logarithmic expressions requires applying the properties of logarithms strategically. By using the power and division rules, we effectively condense complex expressions into concise forms.
Simplifying can reveal equivalence between expressions that might seem different at first glance.
In our exercise, the original expression \( \log_5(x) + 4 \cdot \log_5(y) - 2 \cdot \log_5(z) \) is systematically reduced to \( \log_5\left(\frac{x \cdot y^4}{z^2}\right) \) through these steps.
Understanding each rule and knowing when to apply it allows one to skillfully navigate through problems involving logarithms.
Simplifying can reveal equivalence between expressions that might seem different at first glance.
In our exercise, the original expression \( \log_5(x) + 4 \cdot \log_5(y) - 2 \cdot \log_5(z) \) is systematically reduced to \( \log_5\left(\frac{x \cdot y^4}{z^2}\right) \) through these steps.
Understanding each rule and knowing when to apply it allows one to skillfully navigate through problems involving logarithms.
Other exercises in this chapter
Problem 92
Solve each equation. If necessary, round to the nearest ten-thousandth. $$ \log 4+2 \log x=6 $$
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What is the logarithmic form of the exponential equation \(2^{4}=16 ?\) $$ \mathrm{F} . \log _{16} 2=4 \quad \text { G. } \log _{16} 4=2 \quad \text { H. } \log
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Solve each equation. If necessary, round to the nearest ten-thousandth. $$ 4^{3 x}=77.2 $$
View solution Problem 93
Which function is the inverse of \(f(x)=2^{x-1} ?\) $$ \begin{array}{ll}{\text { A. } f^{-1}(x)=\log _{2}(x-1)} & {\text { B. } f^{-1}(x)=\log _{2}(x+1)} \\ {\t
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