Problem 28 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the logarithm. Round your result to three decimal places.\(\log _{19} 42\)

Step-by-Step Solution

Answer

The value of \( \log_{19} 42 \) rounded to three decimal places is 1.270.

Step 1: Apply the change of base formula

The exercise can be written from base 19 to base 10 using change of base formula as follows: \( \log_{19} 42 = \frac{\log_{10} 42}{\log_{10} 19} \)

Step 2: Calculate individual logarithms

Calculate the individual logarithms using scientific calculator. \n \(\log_{10} 42 \approx 1.62325\) , and \(\log_{10} 19 \approx 1.27875\).

Step 3: Perform the Division

Divide the calculated logarithms. So, \(\frac{1.62325}{1.27875} \) .

Step 4: Round the result

The result from the previous step is approximately 1.26974. Now according to the exercise we need to round the result to three decimal places which will be 1.270.

Key Concepts

Change of Base Formula

Logarithms often come with a variety of bases, like 2, 10, or even 19. But sometimes your calculator might not support those bases directly. That’s where the **Change of Base Formula** comes in handy. It's a special rule that allows us to compute logarithms with one base, using a more convenient base like 10.

Here's the formula:

• If you have a logarithm \(\log_b a\), you can convert it to \(\frac{\log_c a}{\log_c b}\).
• Usually, \(c\) is 10 or \(e\), which are common bases on calculators.

For example, if you want to calculate \(\log_{19} 42\), you would use:

• \(\log_{19} 42 = \frac{\log_{10} 42}{\log_{10} 19}\)
• This makes it possible to solve the logarithm with base 19 with a calculator that only supports base 10.

This formula is very useful, especially during exams, when your toolset might be limited. Don't forget this structure as it will make moving between bases a breeze!

Rounding Numbers

Rounding numbers is all about making a number simpler but still close to what it was. In our exercise, after we compute a division of two logarithms, we need to round the number. Here’s how:

• Imagine you arrive at a number like 1.26974. You want it in a shorter form, such as three decimal places: 1.270.
• Look at the digit just after the place you want to round to. If it’s 5 or more, increase the last kept digit by 1. If less, leave it.

Remember:

• Keeping track of fewer digits is not cheating; it's a standard practice for clarity and neatness in mathematics.
• After rounding, your results will be easier to use in further calculations or when presenting your findings.

Keeping these steps in mind will prevent small errors and give you clean, precise results.

Scientific Calculator Usage

A scientific calculator is your best friend when working with logarithms. It simplifies the process significantly, especially when dealing with complex calculations. Here’s how to effectively use your scientific calculator:

• First, enter the number whose logarithm you want. For example, to find \(\log_{10} 42\), type '42' then press the 'log' button.
• This gives \(1.62325\), which is precise enough for most purposes.
• Next, repeat the process for the base, like '19', and again press 'log'. You should get \(1.27875\).

These steps help you compute each part of the **Change of Base Formula** separately.

Tips for Effective Use:

• Make sure your calculator is in the correct mode to handle these functions.
• Familiarize yourself with the buttons early so you can quickly find logarithms during exams.
• It’s also useful to redo calculations immediately to confirm accuracy.

By mastering these simple processes, you can effortlessly tackle even the daunting logarithm challenges in your studies.