Problem 24 - Intermediate Algebra

Question

For Problems \(23-32\), approximate each of the following logarithms to three decimal places. $$ \log _{3} 32 $$

Step-by-Step Solution

Answer

\( \log_{3} 32 \approx 3.155 \).

Step 1: Use the Change of Base Formula

To approximate the logarithm \( \log_{3} 32 \), we can use the change of base formula: \( \log_{b} a = \frac{\log_{10} a}{\log_{10} b} \). This allows us to convert it to a form that can be easily calculated with a calculator.

Step 2: Calculate the Numerator

Calculate \( \log_{10} 32 \). Using a calculator, we find \( \log_{10} 32 \approx 1.50515 \).

Step 3: Calculate the Denominator

Calculate \( \log_{10} 3 \). Using a calculator, we find \( \log_{10} 3 \approx 0.47712 \).

Step 4: Divide to Find the Result

Now, we perform the division: \( \log_{3} 32 = \frac{\log_{10} 32}{\log_{10} 3} \approx \frac{1.50515}{0.47712} \). This results in approximately \( 3.155 \).

Step 5: Round the Result

Finally, round the result to three decimal places. The result is already at three decimal places, so it remains \( 3.155 \).

Key Concepts

Change of Base Formula

The change of base formula is a powerful tool in mathematics, especially useful when dealing with logarithms that are not in base 10 or base e (natural logarithm). The formula is stated as follows: \( \log_{b} a = \frac{\log_{10} a}{\log_{10} b} \). This allows us to convert a logarithm of any base \(b\) into a form that utilizes common logarithms (base 10), which can be easily calculated using most scientific calculators.
The reason this formula is so useful is that calculators typically only feature buttons for common and natural logarithms. By using the change of base formula, you can evaluate logarithms, such as \( \log_{3} 32 \), without needing a special function for base 3.

• Identify the base \( b \) and the number \( a \) whose logarithm you want to calculate.
• Apply the formula by taking the logarithm of \( a \) and the logarithm of \( b \) using base 10, and then dividing the results.

This approach makes it straightforward and efficient to handle logarithmic expressions with any integer base.

Decimal Approximation

Decimal approximation is the process of finding a number that is close to another number, expressed in decimal form. In the context of logarithms, it's often necessary because

• Logarithms frequently yield irrational numbers.
• These numbers cannot be expressed as a simple fraction, so we approximate them.

To approximate a decimal value, you typically use a calculator, which provides a series of digits in the decimal. For example, in this exercise, \( \log_{10} 32 \approx 1.50515 \) and \( \log_{10} 3 \approx 0.47712 \) were both given as approximations.
After performing necessary calculations, you round the result to a specific number of decimal places. In this task, we aimed for three decimal places, resulting in an approximation of \( 3.155 \).
It's crucial to remember when rounding, you look at the digit following your final place. If it's 5 or greater, you round up. This ensures consistency and accuracy when presenting approximated results.

Mathematical Calculations

Mathematical calculations involving logarithms can seem complex at first, but they follow systematic steps. With tools like calculators, the actual computation becomes manageable, provided you understand each step.
Calculators aid in finding values for logarithmic functions, which can include inputs that yield long decimal sequences or non-terminating numbers. Calculations usually involve:

• Using the change of base formula to translate the logarithm into a calculable form.
• Inputting values into the calculator to derive precise approximations.
• Performing arithmetic like division to consolidate results.

Once the individual logarithmic values (numerator and denominator) have been approximated, you carry out any additional operations required to solve the problem. Lastly, rounding is often the final step, ensuring the result fits the desired precision. Calculations like this prep students for handling real-world data where exact values aren't always possible, promoting familiarity with estimation and approximation techniques in practical applications.