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 \( \log_{16} 57.2 \) to four decimal places is approximately 1.6725.

Step 1: Understanding the properties of logarithms

We can use the change of base formula which states that for any positive number a, b, and x: \( \log_b x = \frac{\log_a x}{\log_a b} \). We can choose either a as base 10 or e. We choose base 10.

Step 2: Change the base

Let's change the base from 16 to 10 using the change of base formula. So, we get: \( \log_{16} 57.2 = \frac{\log_{10} 57.2}{\log_{10} 16} \).

Step 3: Evaluate using a calculator

To finish, use a calculator to evaluate the common logarithms on the right hand side of the equation from Step 2. Remember to round the final result to four decimal places.

Key Concepts

Common Logarithms

Logarithms with base 10 are called common logarithms. We often use "log" to represent them, without specifying the base, because base 10 is frequently used in science and engineering fields.
Common logarithms make calculations easier, especially when using a calculator. If you see a logarithm without a base written next to it, you can safely assume that it's a common logarithm.

• For example, \( \log_{10} 100 = 2 \), since 10 squared is 100.
• Common logarithms can be useful in various calculations, especially in those involving multiple places where it's crucial to have precise values.

When solving for a logarithm like \(\log_{16} 57.2,\) you often involve common logarithms using something called the change of base formula, because most calculators are preset to easily handle these with simple functions.

Natural Logarithms

Natural logarithms use \(e\), which is approximately 2.718, as their base, and they are written as "ln." Natural logarithms are fundamental in higher-level mathematics, particularly in calculus and complex computations.

• For example, \( \ln(e) = 1 \) because the logarithm base \(e\) of \(e\) is 1.
• Natural logarithms are perfect for continuous growth models, like population growth or interest calculations.

Just like common logarithms, natural logarithms can be applied using the change of base formula. However, for simpler arithmetic solutions, you need to convert equations using natural logarithms into a form that can be handled by a standard calculator, which might only directly support common logarithms.

Calculator Usage

Modern calculators typically have functions for evaluating both common and natural logarithms. Here are some tips on how to use your calculator effectively:

• Locate the "log" button for common logarithms and the "ln" button for natural logarithms.
• When using the change of base formula like \(\frac{\log_{10}(57.2)}{\log_{10}(16)}\), enter these sequentially, pressing equals where necessary to ensure accurate results.
• Select either the common logarithm function (base 10) or natural logarithm depending on your application requirements.
• Don't forget to round to the required number of decimal places, such as four decimal places, for precision.

By understanding the functionality of your calculator, you simplify the process and increase both speed and accuracy in solving logarithmic problems.

Properties of Logarithms

Logarithms follow specific rules and properties that make them versatile tools in mathematics. Here are the primary properties you should know:

• **Product Rule**: \(\log_b(MN) = \log_b M + \log_b N\) — this means that the log of a product is the sum of the logs.
• **Quotient Rule**: \(\log_b (\frac{M}{N}) = \log_b M - \log_b N\) — here, the log of a quotient is the difference of the logs.
• **Power Rule**: \(\log_b (M^k) = k \log_b M\) — raising to a power means multiplying the log by that power.
• **Change of Base Formula**: \(\log_b x = \frac{\log_a x}{\log_a b}\) — a useful tool that allows log transformation to any base, making evaluation straightforward when working with calculators.

These properties allow you to manipulate and solve logarithms efficiently, by breaking down complicated expressions into manageable parts. By mastering these properties, you can better approach exercises that involve the evaluation of logarithmic expressions using either common or natural logarithms.