Chapter 6: Problem 36
Use the change-of-base formula to evaluate the logarithm. $$\log _8 22$$
Short Answer
Expert verified
The value of \(\log_8 22\) is approximately \(1.49\) when evaluated using the change-of-base formula.
Step by step solution
01
Understanding the Change of Base Formula
The change-of-base formula translates the logarithm from one base to another. The formula guarantees that the result remains the same. Here it is expressed mathematically: \(\log_b a = \frac{{\log_c a}}{{\log_c b}}\) where \(a\), \(b\), and \(c\) are positive and \(b \neq 1\), \(c \neq 1\)
02
Applying the Change of Base Formula
Apply the change-of-base formula to the given expression \(\log_8 22\). Typically, it's handy to change the base to base \(10\) or base \(e\) for simplicity. Here, we'll change it to base \(10\). So, it becomes: \(\log_8 22 = \frac{{\log_{10} 22}}{{\log_{10} 8}}\)
03
Evaluating the Expression with the New Base
Evaluate the expression \(\frac{{\log_{10} 22}}{{\log_{10} 8}}\). In this case, it's just a matter of punching it into a scientific calculator to get the result.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Logarithms
Logarithms are an essential concept in mathematics, especially when dealing with multiplication and division of large numbers. A logarithm answers the question: To what exponent must we raise a specific base in order to obtain a certain number? For example, if we have the equation \( b^x = y \), then the logarithm of \( y \) with base \( b \) is \( x \), written as \( \log_b(y) = x \).
In practical terms, logarithms are used to simplify complex calculations, such as those involved in exponential growth, decay models, and even sound intensity measurements in decibels. Understanding logarithms can also be helpful in fields like computer science and engineering, where algorithms and calculations often involve exponential factors.
When working with logarithms, it's important to remember a few key properties:
In practical terms, logarithms are used to simplify complex calculations, such as those involved in exponential growth, decay models, and even sound intensity measurements in decibels. Understanding logarithms can also be helpful in fields like computer science and engineering, where algorithms and calculations often involve exponential factors.
When working with logarithms, it's important to remember a few key properties:
- The base of a logarithm must be a positive number and cannot be 1.
- The number we're taking the logarithm of must also be positive.
- Common bases for logarithms are 10 (logarithm base 10) and \( e \) (natural logarithm, written as \( \ln \)).
Evaluating Logarithms
Evaluating logarithms involves finding the power to which a base must be raised to yield a particular number. For example, when you see \( \log_8 22 \), you're being asked 'What power should 8 be raised to in order to get 22?'. This can often be challenging when the numbers do not form a clear power relationship, and that's where the change-of-base formula comes into the picture.
To evaluate a logarithm like \( \log_8 22 \) without a calculator, you'd typically look for powers of 8 that are close to 22. Unfortunately, since there's no integer exponent that will work, we need other methods, such as the change-of-base formula, which allows us to convert to a more convenient base. Using logarithms with base 10 (common logarithms) or base \( e \) (natural logarithms) is particularly useful because these values are preprogrammed into most scientific calculators.
When evaluating logarithms, sometimes you can use properties of logarithms to simplify the expression before calculating. For example, if you have \( \log_{b}(xy) \), you can use the property \( \log_{b}(xy) = \log_{b}(x) + \log_{b}(y) \). These properties make it easier to work with logarithms and solve real-world exponential equations.
To evaluate a logarithm like \( \log_8 22 \) without a calculator, you'd typically look for powers of 8 that are close to 22. Unfortunately, since there's no integer exponent that will work, we need other methods, such as the change-of-base formula, which allows us to convert to a more convenient base. Using logarithms with base 10 (common logarithms) or base \( e \) (natural logarithms) is particularly useful because these values are preprogrammed into most scientific calculators.
When evaluating logarithms, sometimes you can use properties of logarithms to simplify the expression before calculating. For example, if you have \( \log_{b}(xy) \), you can use the property \( \log_{b}(xy) = \log_{b}(x) + \log_{b}(y) \). These properties make it easier to work with logarithms and solve real-world exponential equations.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations that define a particular mathematical relationship. They are fundamental in expressing formulas, solving equations, and modeling real-world scenarios.
When manipulating algebraic expressions, it's crucial to follow the basic principles of algebra, which include the distributive property, associativity, and commutativity. Applying these rules helps to simplify expressions and solve equations effectively. For instance, when dealing with a logarithmic function, you might need to apply these principles to isolate the variable or to convert the expression into a form that can be easily evaluated or simplified.
An understanding of algebraic expressions goes hand in hand with logarithms because the properties of logs are applied within the algebraic framework. Whether it's expanding, factoring, or simplifying expressions, a strong grasp of algebra is necessary for handling logarithmic equations and making sense of their solutions. Through practice and application of these algebraic principles, students can solve a wide range of problems more efficiently, translating theoretical knowledge into practical skills.
When manipulating algebraic expressions, it's crucial to follow the basic principles of algebra, which include the distributive property, associativity, and commutativity. Applying these rules helps to simplify expressions and solve equations effectively. For instance, when dealing with a logarithmic function, you might need to apply these principles to isolate the variable or to convert the expression into a form that can be easily evaluated or simplified.
An understanding of algebraic expressions goes hand in hand with logarithms because the properties of logs are applied within the algebraic framework. Whether it's expanding, factoring, or simplifying expressions, a strong grasp of algebra is necessary for handling logarithmic equations and making sense of their solutions. Through practice and application of these algebraic principles, students can solve a wide range of problems more efficiently, translating theoretical knowledge into practical skills.
