Problem 17
Question
Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log N^{-6}\)
Step-by-Step Solution
Verified Answer
The expanded expression of \(\log N^{-6}\) is -6\(\log N\).
1Step 1: Use the power property of logarithms
The power property of logarithms states that \(\log a^{b} = b \cdot \log a\). Applying this property, \(\log N^{-6}\) becomes -6 \(\log N\).
2Step 2: Final answer
There is no further simplification necessary as we can't evaluate \(\log N\) without knowing \(N\). Therefore, the expanded expression is -6 \(\log N\).
Key Concepts
Properties of LogarithmsPower PropertyExpanding Logarithms
Properties of Logarithms
When working with logarithmic expressions, it's important to understand their properties. These properties help us simplify and expand expressions, making them easier to work with or solve. Here are some key properties you should remember:
- **Product Property**: This states that the logarithm of a product is the sum of the logarithms of the factors. In mathematical terms: \( \log(ab) = \log a + \log b \).
- **Quotient Property**: This involves the division of numbers inside a logarithm. The property states that the logarithm of a quotient is the difference of the logarithms: \( \log \left( \frac{a}{b} \right) = \log a - \log b \).
- **Power Property**: This is particularly useful in our given problem. It states that the logarithm of a power can be simplified by multiplying the exponent by the logarithm of the base: \( \log a^b = b \cdot \log a \).
Power Property
The power property of logarithms is especially handy when you have a logarithmic expression with an exponent. It is a clear and straightforward tool for simplification.
The power property states that \[\log a^b = b \cdot \log a\]This means that if you have a number raised to a power inside a logarithm, you can "move" the exponent out front, multiplying it by the logarithm itself. This simplifies the expression, often making it easier to handle.
In our exercise, we applied this property to \[\log N^{-6}\] By using the power property, the expression becomes:\[\log N^{-6} = -6 \cdot \log N\]This step effectively expanded the logarithm by removing the negative exponent and multiplying it outside the logarithmic function. This simplification is useful because it makes complex logarithmic operations more straightforward and facilitates further calculations.
The power property states that \[\log a^b = b \cdot \log a\]This means that if you have a number raised to a power inside a logarithm, you can "move" the exponent out front, multiplying it by the logarithm itself. This simplifies the expression, often making it easier to handle.
In our exercise, we applied this property to \[\log N^{-6}\] By using the power property, the expression becomes:\[\log N^{-6} = -6 \cdot \log N\]This step effectively expanded the logarithm by removing the negative exponent and multiplying it outside the logarithmic function. This simplification is useful because it makes complex logarithmic operations more straightforward and facilitates further calculations.
Expanding Logarithms
Expanding logarithms refers to the process of using logarithmic properties to rewrite a log expression in a more detailed form. This process makes certain operations more manageable, especially when dealing with complex expressions.
The goal is to break down the logarithmic expressions using properties like the product, quotient, and power properties, as introduced earlier. For instance, turning a product or quotient into a sum or difference of logs; and converting a power into a multiplier.
In the exercise, you expanded \( \log N^{-6} \) using the power property. This was necessary because you had an expression with a power. The result was:\[-6 \cdot \log N\]This is as expanded as the expression can get without additional information about \( N \). The advantage of expanding is seen in simplifying the arithmetic operations or when working within larger expressions, making each individual part easier to compute or understand. Expanding is crucial when you need to compare, solve for variables or further analyze logarithmic equations.
The goal is to break down the logarithmic expressions using properties like the product, quotient, and power properties, as introduced earlier. For instance, turning a product or quotient into a sum or difference of logs; and converting a power into a multiplier.
In the exercise, you expanded \( \log N^{-6} \) using the power property. This was necessary because you had an expression with a power. The result was:\[-6 \cdot \log N\]This is as expanded as the expression can get without additional information about \( N \). The advantage of expanding is seen in simplifying the arithmetic operations or when working within larger expressions, making each individual part easier to compute or understand. Expanding is crucial when you need to compare, solve for variables or further analyze logarithmic equations.
Other exercises in this chapter
Problem 17
Write each equation in its equivalent logarithmic form. $$b^{3}=1000$$
View solution Problem 17
Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. $$f(x)=(0.6)^{x}$$
View solution Problem 18
The half-life of the radioactive element plutonium-239 is \(25,000\) years. If 16 grams of plutonium- 239 are initially present, how many grams are present afte
View solution Problem 18
Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$9^{x}=\frac{1}{\sqrt[3]{3}}$$
View solution