Problem 67 - Algebra 2

Question

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. \(\log _{4} 68\)

Step-by-Step Solution

Answer

\( \log_4 68 \approx 3.0422 \)

Step 1: Change of Base Formula

The change of base formula allows us to express a logarithm in terms of logs of any base. The formula is given by \( \log_b a = \frac{\log_c a}{\log_c b} \). We can use the common logarithm where the base is 10, denoted as \( \log \). Thus, \( \log_4 68 = \frac{\log 68}{\log 4} \).

Step 2: Calculate Common Logarithms

Compute the common logarithms using a calculator: \( \log 68 \) and \( \log 4 \). For approximation purposes, \( \log 68 \approx 1.8325 \) and \( \log 4 \approx 0.6021 \).

Step 3: Apply the Change of Base Formula

Substitute the approximated values into the change of base formula: \( \log_4 68 \approx \frac{1.8325}{0.6021} \).

Step 4: Simplify the Expression

Divide the values to approximate the expression: \( \frac{1.8325}{0.6021} \approx 3.0422 \).

Step 5: Conclusion

Thus, \( \log_4 68 \) approximates to \( 3.0422 \) when calculated using common logarithms and rounding to four decimal places.

Key Concepts

Common Logarithms

Common logarithms are logarithms with base 10. They are often used in various fields of science and engineering because of their simplicity and convenience. When you see a logarithm written without a base, such as \( \log x \), it is typically assumed to be a common logarithm. This means the base is 10.

• The notation \( \log x \) implies \( \log_{10} x \).
• Common logarithms are easy to use with calculators, as most have a dedicated button for \( \log \).

Using common logarithms can simplify problems significantly, especially when employing the change of base formula, as it allows calculations involving logarithms with different bases to be performed using base 10.

Logarithmic Approximation

Logarithmic approximation is the process of estimating logarithmic values using a calculator or a logarithm table. Exact values can be quite complex to calculate manually, which is why approximation is essential. This method is useful when dealing with non-integer bases or arguments. For the example given, we calculate:

• \( \log 68 \approx 1.8325 \)
• \( \log 4 \approx 0.6021 \)

These values are derived from a scientific calculator or logarithmic tables, which provide logarithmic values to multiple decimal places. Approximations are typically rounded to a desired number of decimal places, four in our case, to ensure precision while keeping calculations manageable.

Base 10 Logarithms

Base 10 logarithms, also known as common logarithms, are widely used and recognized due to their historical significance and ease of use. They allow us to convert multiplication into addition, division into subtraction, powers into multiplication, and roots into division. These properties make them very powerful tools in mathematics.

• Base 10 is considered standard in many applications.
• Calculators and computer algorithms are optimized to handle base 10 logarithms efficiently.

When converting logarithms of other bases via the change of base formula, base 10 logarithms often serve as the go-to choice, providing straightforward interpretation and calculation using readily available tools.