Problem 55 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\ln 28^{3}$$

Step-by-Step Solution

Answer

\(\ln 28^3 \approx 9.9966\).

Step 1: Simplify the Logarithm Expression

The expression provided is \(\ln 28^3\). First, use the power rule of logarithms, which states that \(\ln(a^b) = b \cdot \ln(a)\). Applying this rule gives us \(3 \cdot \ln 28\).

Step 2: Calculate \(\ln 28\) Using a Calculator

Use a calculator to find \(\ln 28\). Enter 28 into the calculator and press the natural logarithm (ln) function key to get its value. You will get approximately \(\ln 28 \approx 3.3322\).

Step 3: Multiply by 3

Now, multiply the result from the previous step by 3 to calculate \(3 \cdot 3.3322\). Perform the multiplication: \(3 \times 3.3322 = 9.9966\).

Step 4: Round the Result

Typically, you round the result to a certain decimal place for simplicity. In most cases, rounding to four decimal places is sufficient. Therefore, the rounded result is approximately 9.9966.

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a handy tool when dealing with logarithmic expressions that contain exponents. This rule states that for any logarithm of the form \(\ln(a^b)\), it can be simplified to \(b \cdot \ln(a)\).
This means that you can bring the exponent out front as a multiplier. In our example, \(\ln 28^3\) becomes \(3 \cdot \ln 28\) using this rule.
Applying this rule simplifies calculations and helps to manage large numbers more easily. Remember, the base of the logarithm must be positive and not equal to 1 for this rule to apply.

Decimal Approximation

Decimal approximation is used to convert the values of logarithms, which often result in irrational numbers, into a practical and usable form. When computing \(\ln 28\), a calculator gives us approximately 3.3322. This approximation is essential because exact values of many logarithms cannot be written in decimal form.

• Always use a calculator capable of computing natural logarithms for accurate approximations.
• Rounding is crucial after calculation, and typically it is appropriate to round to four decimal places, ensuring both precision and simplicity.

Thus, the expression \(3 \cdot 3.3322\) is approximately 9.9966, showcasing how decimal approximation makes complex expressions manageable by providing clear, concise results.

Logarithmic Calculators

Logarithmic calculators are powerful tools designed to handle the calculation of logarithmic expressions efficiently. These calculators come in various forms such as physical calculators, online tools, or software applications.
To compute the natural logarithm \(\ln 28\), simply input 28 and use the ln key on the calculator. It quickly provides a decimal representation, enabling further calculations or approximations.

• Ensure your calculator is set to the correct mode, focusing on natural logarithms rather than common logarithms which are base 10.
• Understand the limitations and settings of your calculator to avoid mode errors and gain accurate results.

Leveraging logarithmic calculators simplifies the mathematical process, making even complex logarithmic expressions easy to solve and understand.