Problem 45 - Precalculus : Building Concepts and Connections

Question

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{5} 0.5$$

Step-by-Step Solution

Answer

-0.4307

Step 1: Write the change of base formula

The change of base formula is given by: \(\log_b a = \log_d a / \log_d b\). Our logarithmic expression can be written as \(\log _{5} 0.5 = \log _{10} 0.5 / \log _{10} 5\). This is because we are changing the base from 5 to 10.

Step 2: Evaluate the logarithms

Using a calculator, \(\log _{10} 0.5\) is found to be -0.3010 and \(\log _{10} 5\) is found to be 0.6989.

Step 3: Perform the division

The result can now be found by dividing the two values calculated above: -0.3010 / 0.6989.

Step 4: Round the answer

Lastly, the answer must round to four decimal places. So, apply the round operation to get the final answer.

Key Concepts

Logarithmic Expressions

Logarithmic expressions are mathematical statements that involve logarithms, which are the inverses of exponential functions. A logarithm answers the question: “To what exponent must the base be raised to obtain a certain number?”. In the expression \( \log_b a \), \( b \) is called the base, and \( a \) is the number you are finding the logarithm of.

The change of base formula is a key tool when working with logarithmic expressions, especially when you need to evaluate a log that isn't to a base your calculator readily supports. The change of base formula is defined as:

• \( \log_b a = \frac{\log_d a}{\log_d b} \)

By using a convenient base like 10 or \( e \) (natural log), you can transform difficult-to-evaluate logs into something more manageable. This is particularly useful when calculators are involved because most can handle base 10 (common logarithm) and base \( e \) (natural logarithm) easily.

Calculator Usage

Using a calculator effectively is crucial when working with logarithmic functions, especially when applying the change of base formula. Not all calculators have a function for bases other than 10 or \( e \). Thus, master the change of base technique to handle logarithms with any base.

When using a calculator, follow these steps:

• Identify the logarithmic expression and apply the change of base formula.
• Input \( \log_{10} \text{(the number)} \) and \( \log_{10} \text{(the base)} \) separately.
• Ensure to enter values carefully to avoid errors.

After performing these calculations, it's beneficial to understand what your results signify. Here, \( \log_{10} 0.5 \) returns about -0.3010 and \( \log_{10} 5 \) provides approximately 0.6989. Performing the division gives you the value of the original log expression, in this case, \( \frac{-0.3010}{0.6989} \).

Getting familiar with this process not only builds confidence but also enhances your numerical reasoning as you gain insight into manipulating exponential relationships effectively.

Rounding Decimals

Rounding decimals is a common practice in math to make numbers easier to work with, especially when an exact answer is not necessary or when a solution needs to be presented within a certain level of precision. It is essential when reporting answers in mathematics, ensuring clarity and simplicity.

When rounding to four decimal places:

• Look at the fifth decimal digit to determine whether to round up or stay the same.
• If the fifth digit is 5 or more, add one to the fourth digit.
• If the fifth digit is less than 5, leave the fourth digit as is.

In our logarithmic example, after dividing -0.3010 by 0.6989, you may obtain a long decimal. The practice of rounding ensures the answer is manageable, typically aligning it with what is required for school assignments or data presentations. Applying rounding allows us to say that \( \log_5 0.5 \) is approximately -0.4303 when rounded to four decimal places.