Problem 74 - College Algebra

Question

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{16} 57.2 $$

Step-by-Step Solution

Answer

The value of \(\log_{16}{57.2}\), calculated to four decimal places, can be calculated using a calculator and the formula for changing base of logarithms.

Step 1: Express the logarithm base in terms of common or natural base

Use the formula for changing base of logarithm. The formula is: \(\log_{b}{a} = \frac{\log_{c}{a}}{\log_{c}{b}}\). In the case of the given exercise, 'b' is 16 and 'a' is 57.2. You can choose 'c' as 10 (for common log) or 'e' (for natural log). Since the exercise does not specify, either is acceptable. If we choose 'c' as 10, the equation becomes: \(\log_{16}{57.2} = \frac{\log_{10}{57.2}}{\log_{10}{16}}\).

Step 2: Use a calculator to evaluate the expression

Use a calculator to compute the values of \(\log_{10}{57.2}\) and \(\log_{10}{16}\) to four decimal places.

Step 3: Compute the division

Divide the value obtained for \(\log_{10}{57.2}\) by the value obtained for \(\log_{10}{16}\) to get the value of \(\log_{16}{57.2}\).

Key Concepts

Understanding Change of Base Formula

The change of base formula is a handy tool in mathematics when dealing with logarithms that do not have a base commonly found on calculators, such as 10 or \(e\). It's a method to convert logarithms into any new base that is more convenient to use. The formula is \(\log_{b}{a} = \frac{\log_{c}{a}}{\log_{c}{b}}\), where \(b\) is the base of the logarithm you have, \(a\) is the argument (the number you are taking the log of), and \(c\) is the new base you want to convert to.
For example, if you are given \(\log_{16}{57.2}\), converting it to base 10 using this formula would be \(\frac{\log_{10}{57.2}}{\log_{10}{16}}\). This transformation is particularly useful because most calculators have built-in functions for common logarithms (base 10) and natural logarithms (base \(e\)). This means you can evaluate logarithms that originally have a base of 16, or any other number, using simple algebra and computational tools.

Exploring Common Logarithms

Common logarithms are logarithms with the base 10. They are often written as \(\log_{10}{x}\) or simply \(\log{x}\). These are widely used in scientific calculations because they simplify the scaling of calculations and expressions involving powers of ten, which are very common in these fields.
When dealing with base conversions in logarithms, using common logarithms can make calculations more straightforward thanks to the prevalence of calculators that can perform these operations. In the example \(\log_{16}{57.2}\), using common logarithms simplifies the calculation to \(\frac{\log_{10}{57.2}}{\log_{10}{16}}\). This straightforward approach avoids complications and expedites solving problems that may involve non-standard bases.

• Useful for solving exponential equations that involve powers of 10.
• Commonly available in calculators, facilitating easy computation.

Understanding common logarithms is essential, as they offer a standardized approach for many real-world mathematical applications.

Introduction to Natural Logarithms

Natural logarithms have a special base \(e\), which is an irrational number approximately equal to \(2.71828\). Represented as \(\ln(x)\), natural logarithms arise naturally in many mathematical contexts, particularly those involving growth processes like population growth, radioactive decay, and compound interest.
The natural logarithm is notated as \(\log_e{x}\) but is commonly written as \(\ln{x}\), and it is significant because it relates to exponential functions which frequently appear in continuous processes.

• The constant \(e\) is fundamental in mathematics, especially in calculus.
• Natural logarithms simplify the understanding of growth rates.

While the exercise \(\log_{16}{57.2}\) allows for evaluation using either common or natural logarithms due to the change of base formula, understanding the concept of natural logarithms enriches your capacity to apply logarithms to both theoretical and practical problems.