Problem 6
Question
Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log (10,000 x)\)
Step-by-Step Solution
Verified Answer
The expanded and simplified form of the logarithmic expression \(\log (10,000x)\) is \(4 + \log(x)\)
1Step 1: Apply the product rule of logarithms
The product rule for logarithms states that \(\log(ab) = \log(a) + \log(b)\). As such, we start by applying this rule to \(\log (10,000x)\), which gives \(\log(10,000) + \log(x)\).
2Step 2: Evaluate the logarithm
We know that \(\log(10,000)\) is equal to 4 because \(10^4\) is \(10,000\). Hence, the expression simplifies down to \(4 + \log(x)\).
3Step 3: Write the final simplified expression
The logarithmic expression has been completely expanded and simplified. The final answer is \(4 + \log(x)\)
Key Concepts
Product Rule of LogarithmsEvaluating LogarithmsLogarithmic Expressions
Product Rule of Logarithms
Understanding the product rule of logarithms is essential for expanding and simplifying logarithmic expressions. It states that the logarithm of a product is equal to the sum of the logarithms of the factors. Mathematically, this is represented as \( \log(ab) = \log(a) + \log(b) \). This property is derived from the exponential form, reflecting how multiplying numbers translates into adding their exponents.
For example, when given \( \log(10,000x) \) to expand, we treat \(10,000 \) and \(x\) as separate factors of the product. Applying the product rule gives us two separate logarithms: \(\log(10,000) + \log(x)\). This decomposition is a crucial step for simplifying the expression, especially when one of the factors is a known base power, allowing for further evaluation without a calculator. Knowing this property helps in breaking down complex logarithmic expressions into more manageable parts.
For example, when given \( \log(10,000x) \) to expand, we treat \(10,000 \) and \(x\) as separate factors of the product. Applying the product rule gives us two separate logarithms: \(\log(10,000) + \log(x)\). This decomposition is a crucial step for simplifying the expression, especially when one of the factors is a known base power, allowing for further evaluation without a calculator. Knowing this property helps in breaking down complex logarithmic expressions into more manageable parts.
Evaluating Logarithms
Evaluating logarithms without a calculator is a valuable skill that relies on understanding the relationship between logarithms and exponents. The logarithm \(\log_b(a)\) answers the question: 'To what exponent must the base \(b\) be raised to produce \(a\)?' For instance, when we have \(\log(10,000)\), we determine the power that 10 must be raised to, which is 4, because \(10^4 = 10,000\).
Through practice and familiarity with common base powers, one can quickly evaluate simple logarithms. Moreover, this skill also helps identify when logarithmic expressions can be simplified without further expansion, streamlining the process of working through logarithmic problems. It's important to recognize patterns and remember that \(\log_b(b) = 1\) and \(\log_b(1) = 0\), as these are often used as stepping stones in the evaluating process.
Through practice and familiarity with common base powers, one can quickly evaluate simple logarithms. Moreover, this skill also helps identify when logarithmic expressions can be simplified without further expansion, streamlining the process of working through logarithmic problems. It's important to recognize patterns and remember that \(\log_b(b) = 1\) and \(\log_b(1) = 0\), as these are often used as stepping stones in the evaluating process.
Logarithmic Expressions
Logarithmic expressions represent the logarithms of numbers, variables, or expressions. Simplifying or expanding logarithmic expressions involves applying properties of logarithms, like the product rule, quotient rule, and power rule. When approaching a problem such as \(\log (10,000x)\), we first identify any properties that can be used.
Once expanded using the appropriate logarithm rules—such as the product rule in this case—we aim to evaluate parts of the expression that correspond to known values or can be easily calculated. In our example, \(\log (10,000)\) could be directly evaluated because it corresponds to a base power that is commonly known. As a result, the original expression becomes easier to understand and use in subsequent calculations.
Understanding how to rewrite and manipulate these expressions is critical in higher mathematics, where logarithms play a central role in solving exponential equations and working with growth and decay problems.
Once expanded using the appropriate logarithm rules—such as the product rule in this case—we aim to evaluate parts of the expression that correspond to known values or can be easily calculated. In our example, \(\log (10,000)\) could be directly evaluated because it corresponds to a base power that is commonly known. As a result, the original expression becomes easier to understand and use in subsequent calculations.
Understanding how to rewrite and manipulate these expressions is critical in higher mathematics, where logarithms play a central role in solving exponential equations and working with growth and decay problems.
Other exercises in this chapter
Problem 5
Approximate each number using a calculator. Round your answer to three decimal places. $$4^{-1.5}$$
View solution Problem 6
Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$3^{2 x+1}=27$$
View solution Problem 6
Write each equation in its equivalent exponential form. $$3=\log _{b} 27$$
View solution Problem 6
Approximate each number using a calculator. Round your answer to three decimal places. $$6^{-1.2}$$
View solution