Question

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{2 / 3} 32 $$

Short answer

The solution to the problem is approximately \(-6.726\).

Step-by-step solution

Apply the Change of Base Formula

Use the change-of-base formula to rewrite the expression \(\log _{2 / 3} 32\). In this case, take \(a = 32\), \(b = 2/3\), and \(d\) any positive number usually base \(10\) or \(e\). For standard calculators, \(d\) is most commonly base \(10\). So, the equation now becomes: \(\log_{10} 32 / \log_{10} (2/3)\).

Use Calculator to Evaluate Logarithms

Use a calculator to evaluate \(\log_{10} 32\) and \(\log_{10} (2/3)\).

Divide the Results

Divide the result of \(\log_{10} 32\) by the result of \(\log_{10} (2/3)\). Round the outcome to three decimal places as the problem requests.

Key concepts

Evaluating Logarithms

Understanding the concept of logarithms is crucial to solving various algebraic problems. A logarithm, essentially, is the opposite of exponentiation. It answers the question: To what power must we raise a given base to obtain a certain number? The general form of a logarithm is \( \log_b a \) which asks for the power \( x \) so that \( b^x = a \).

To evaluate logarithms without a calculator, one would typically need to rely on the properties of logarithms or recognize standard values. However, for non-standard values such as \( \log _{2 / 3} 32 \), using a calculator is a practical necessity. The step-by-step solution correctly suggests starting with the change-of-base formula, which allows us to convert the base of a logarithm to something more calculator-friendly, typically \( \log_{10} \) or \( \ln \), the natural logarithm. This powerful tool ensures that we can tackle logarithms with any base by using the common logarithms our calculators are designed to handle.

Calculator Use in Algebra

In algebra, calculators serve as an indispensable tool, not just for simple arithmetic, but also for complex operations like evaluating logarithms. With modern scientific calculators, tasks that were once long and tedious can now be completed in a fraction of the time.

When approaching a problem like \( \log _{2 / 3} 32 \), which involves a non-standard base, calculators help us employ the change-of-base formula. To effectively use a calculator in this context, we input the values into the calculator after applying the formula. For example, one would separately calculate \( \log_{10} 32 \) and \( \log_{10} (2/3) \) and then perform the division. For complex expressions, it's essential to understand the order of operations and parentheses usage to ensure the calculator interprets your input correctly. Always refer to your calculator's manual for specific functions and capabilities, as these can dramatically simplify algebraic problem-solving.

Rounding Decimal Places

In mathematics, the concept of rounding is used to adjust numbers to make them simpler and more manageable, often to match the level of precision required by the context of the problem. Rounding to a specific number of decimal places is a common request in algebra problems.

After using a calculator to find the value of \( \log_{10} 32 \) and \( \log_{10} (2/3) \), dividing them, as we did in our example problem, will often give a long decimal result. Rounding is necessary for clarity and simplicity. In the exercise, we are asked to round the final answer to three decimal places, which means the answer should have only three numbers to the right of the decimal point. It's important to pay attention to the fourth digit—if it's five or above, you round up the third digit; if it's four or below, you keep the third digit as is. Accurate rounding affects the final answer significantly, especially in fields that rely on precision, like science and engineering.