Problem 23 - Intermediate Algebra

Question

For Problems \(23-32\), approximate each of the following logarithms to three decimal places. $$ \log _{2} 23 $$

Step-by-Step Solution

Answer

\(\log_{2} 23 \approx 4.526\)

Step 1: Change of Base Formula

To approximate \(\log_{2} 23\), we use the change of base formula: \(\log_{b} a = \frac{\log_{c} a}{\log_{c} b}\). We will use base 10 for convenience. So, \(\log_{2} 23 = \frac{\log_{10} 23}{\log_{10} 2}\).

Step 2: Calculate \(\log_{10} 23\)

Using a calculator, compute \(\log_{10} 23\). The value of \(\log_{10} 23\) is approximately equal to 1.362.

Step 3: Calculate \(\log_{10} 2\)

Again, using a calculator, find \(\log_{10} 2\). The value of \(\log_{10} 2\) is approximately equal to 0.301.

Step 4: Divide the Results

Now, divide the result from Step 2 by the result from Step 3: \(\frac{1.362}{0.301} \approx 4.525\).

Step 5: Round to Three Decimal Places

Finally, round 4.525 to three decimal places. Hence, \(\log_{2} 23 \approx 4.526\).

Key Concepts

Change of Base Formula

When working with logarithms that aren't in the familiar base 10 or natural logarithms, we often use the change of base formula to simplify calculations. This formula is a powerful tool for converting logarithms from one base to another, making computations easier. It states:

• For any numbers \(a\), \(b\), and a new base \(c\), the formula is given by \(\log_{b} a = \frac{\log_{c} a}{\log_{c} b}\).
• It allows us to convert any logarithm into a more manageable form, particularly when dealing with calculators that handle base 10.

In our problem, to approximate \(\log_{2} 23\), we used base 10. This transforms our original logarithmic expression to \(\frac{\log_{10} 23}{\log_{10} 2}\). This approach simplifies the process and enables us to use a regular calculator to find each individual logarithm value and then solve the equation through division.

Base 10 Logarithm

Base 10 logarithms, often referred to as common logarithms, are commonly used in many mathematical and scientific calculations. They are represented as \(\log_{10}\) or simply \(\log\) when the base is known from context.

• They are particularly useful because calculations are simplified using typical calculators, which are designed to compute base 10.
• In cases like our example, where we need to compute \(\log_{10} 23\), the calculator directly provides an approximate value, beneficial for manual computation.

In the step-by-step solution, we identified \(\log_{10} 23\) as approximately 1.362. Understanding the operation of base 10 logs can make solving similar problems much less daunting.

Approximating Logarithms

Approximating logarithms involves calculating logarithm values to a predetermined degree of accuracy, often three decimal places for simplicity.

• This level of precision is usually sufficient for quick hand calculations or when an exact number isn't required.
• The approximation is typically derived using calculators that compute and provide decimal representations of logarithms quickly.

In mathematical exercises like this one, we approximate \(\log_{10} 2\) as well, with a calculated value close to 0.301. This helps us understand the initial estimates of logarithms needed for accurate division and further calculations.

Mathematical Calculations

The final step in using the change of base formula and the computed logs is performing the division and rounding off the result. These mathematical calculations, though seemingly simple, are vital:

• After obtaining the values \(\log_{10} 23 \approx 1.362\) and \(\log_{10} 2 \approx 0.301\), divide them to find \(\log_{2} 23\).
• The division yields a number that needs the precision of rounding—hence, \(\frac{1.362}{0.301} = 4.525\), which rounds to three decimal places as 4.526.

These steps underscore not only familiarity with calculations but attention to rounding, which ensures the precision expected in most academic and practical applications.