Problem 67 - College Algebra with Modeling and Visualization

Question

Use the change of base formula to approximate the logarithm to the nearest thousandth. $$\log _{2} 25$$

Step-by-Step Solution

Answer

The approximate value of \(\log_2 25\) is 4.644.

Step 1: State the Change of Base Formula

The change of base formula for logarithms is used to convert logarithms from one base to another. The formula is: \[ \log_b a = \frac{\log_c a}{\log_c b} \] where \(b\) is the original base, \(a\) is the number you are taking the logarithm of, and \(c\) is the new base. Typically, \(c\) is chosen to be 10 or \(e\) to use calculators easily.

Step 2: Apply the Change of Base Formula

For the given problem \(\log_2 25\), we will use the change of base formula with base 10: \[ \log_2 25 = \frac{\log_{10} 25}{\log_{10} 2} \].

Step 3: Calculate \(\log_{10} 25\) and \(\log_{10} 2\)

Using a calculator, find \(\log_{10} 25\):\(\log_{10} 25 \approx 1.39794\), and \(\log_{10} 2 \approx 0.30103\).

Step 4: Divide to Find \(\log_{2} 25\)

Divide the results from Step 3:\[ \log_2 25 \approx \frac{1.39794}{0.30103} \approx 4.64385 \].

Step 5: Round to the Nearest Thousandth

Round the result from Step 4 to the nearest thousandth to find the final answer:\[ \log_2 25 \approx 4.644 \].

Key Concepts

Understanding Logarithms

A logarithm answers the question, "To what power must we raise a certain number, called the base, to obtain another number?" When you see something like \( \log_b a \), it means "what power do we raise \( b \) to in order to get \( a \)?" Logarithms are the inverse operations of exponentiation. They are useful in many areas of science and engineering where exponential growth or decay occurs.
For example, in \( \log_2 25 \), the base is 2 and we want to discover what power of 2 equals 25. This is not an integer, so we use a technique like the change of base formula to calculate it.

• Basics of logarithms: It involves a base and a number.
• Representation: \( \log_b a = c \) means \( b^c = a \).

Base Conversion Using Change of Base Formula

Base conversion in logarithms involves changing the base of a logarithm to make calculations easier. The change of base formula is a pivotal tool for this. It allows for the computation of logarithms with bases that are challenging to work with directly by converting them into logarithms of bases that are more convenient.
The formula \( \log_b a = \frac{\log_c a}{\log_c b} \) is a classic example. Here, you select a new base \( c \), usually 10 or \( e \), because calculators handle these bases well. So, to compute \( \log_2 25 \), we use base 10:
\[ \log_2 25 = \frac{\log_{10} 25}{\log_{10} 2} \]
Explore this formula to convert any complex logarithmic expression into a more calculable form.

The Role of Mathematical Approximation

Mathematical approximation is crucial when working with logarithms, especially when the exact value is not easily obtainable. By using a calculator, we can approximate the logarithmic values needed for base conversions. It’s similar to estimating when you can't find an exact result quickly.
For \( \log_2 25 \), we approximate \( \log_{10} 25 \approx 1.39794 \) and \( \log_{10} 2 \approx 0.30103 \). These are obtained from calculators as they don't result in tidy whole numbers.

• Purpose: Approximate values to enable further calculation.
• Precision: Approximations are often rounded to a convenient decimal place, like to the nearest thousandth.

This rounding helps achieve a practical level of accuracy in everyday calculations.