Problem 34 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{5} 12 $$

Step-by-Step Solution

Answer

The value of \( \log_{5} 12 \) rounded to three decimal places is approximately 1.543.

Step 1: Enter the function into the calculator

Start by entering the logarithmic function \( \log_{5} 12 \) into a scientific calculator. It's important to ensure that the base of the logarithm (which is 5 in this case) and the number 12 are entered correctly.

Step 2: Calculate using the calculator

Most calculators will instantly calculate the value. In this case, the calculator will perform the calculation and display a result.

Step 3: Round the result

Once the calculator has returned a result, it needs to be rounded to three decimal places. If the fourth decimal is 5 or above, round the third decimal up. If it's 4 or lower, keep the third decimal as is.

Key Concepts

Scientific Calculator Usage

Using a scientific calculator can be straightforward if you are aware of some basic functions. Firstly, recognize that not all scientific calculators have a direct button for custom-base logarithms like \( \log_5 \). Most commonly, calculators provide the base 10 log, labeled as "log", and the natural log, labeled as "ln". To calculate \( \log_5 12 \), you typically need to use the change of base formula, which involves calculating \( \log_{10} 12 \) divided by \( \log_{10} 5 \). You would enter it in the calculator like this:

• Press the "log" button for base 10 log.
• Enter 12, then close the parenthesis and divide by "log", followed by 5.
• Complete with the equal sign to get the final value.

After inputting this correctly, your calculator gives you a precise and rapid result, ready for rounding.

Rounding Decimals

Rounding decimals is often necessary in mathematics to provide an easier-to-read and precise enough answer. When rounding to three decimal places, you look at the fourth decimal to decide what to do with the third. Here’s how it works:

• If the fourth decimal is 5 or higher, increase the third decimal by 1.
• If the fourth decimal is 4 or lower, keep the third decimal unchanged.

For example, if your calculator shows 1.5447, look at the last digit, 7. Since 7 is greater than 5, you would round up the third digit, making the number 1.545.
Practice makes perfect. Every time you round, it ensures that your results are both accurate and understandable.

Base Conversion

The concept of base conversion in logarithms emerges, especially when using a calculator that doesn't support custom bases. As demonstrated earlier, you can use the change of base formula to tackle this. The change of base formula is crucial because calculators typically only handle base 10 or natural logarithms directly.
Let's say the logarithm is \( \log_b a \), and your calculator only does \( \log_{10} \). Then:

• Compute \( \log_b a \) by \( \frac{\log_{10} a}{\log_{10} b} \)
• This simplifies the problem so you can engage directly with the calculator's standard functions.

Base conversion allows flexibility beyond calculator limitations, giving you access to a wider range of problems without being constrained by a calculator’s specific functions.