Problem 33 - Algebra and Trigonometry

Question

\(33-36\) Use a calculator to evaluate the expression, correct to four decimal places. $$ \begin{array}{llll}{\text { (a) } \log 2} & {\text { (b) } \log 35.2} & {} & {\text { (c) } \log \left(\frac{2}{3}\right)}\end{array} $$

Step-by-Step Solution

Answer

(a) 0.3010, (b) 1.5478, (c) -0.1761

Step 1: Calculate \( \log 2 \)

Use a calculator to find the logarithm of 2. Input the number into your calculator and press the log button to find the result.\(\log 2 \approx 0.3010\)

Step 2: Calculate \( \log 35.2 \)

Use your calculator to find the logarithm of 35.2. Enter 35.2 and press the log function.\(\log 35.2 \approx 1.5478\)

Step 3: Calculate \( \log \left(\frac{2}{3}\right) \)

First, find the value of \( \frac{2}{3} \), which is approximately 0.6667. Now, use your calculator to calculate the logarithm of this number.\(\log \left(\frac{2}{3}\right) \approx -0.1761\)

Key Concepts

Calculator Usage

When it comes to evaluating logarithms, using a calculator can simplify the process greatly. Calculators have built-in functions to handle logarithms, which makes finding these values quick and accurate.
To evaluate a logarithm like \(\log 2\), simply enter the number 2 into your calculator first. Then, press the button labeled "log" or sometimes "\(\log_{10}\)" to display the logarithm base 10 of the number entered. This will give you the result \(\log 2 \approx 0.3010\).
For more complex numbers like \(\log 35.2\), the process is the same. Just enter 35.2 followed by pressing the "log" button, resulting in \(\log 35.2 \approx 1.5478\). Calculators process this operation using pre-programmed algorithms, ensuring precision up to several decimal places. Using these steps will help you evaluate any logarithm you encounter with ease!

Logarithm Properties

Understanding the properties of logarithms is crucial when dealing with more complex expressions. There are main properties or laws that are particularly helpful:

• Product Property: \(\log(a \cdot b) = \log a + \log b\)
• Quotient Property: \(\log\left(\frac{a}{b}\right) = \log a - \log b\)
• Power Property: \(\log(a^b) = b \cdot \log a\)

For instance, when dealing with \(\log\left(\frac{2}{3}\right)\), the quotient property is quite handy. Though you can directly compute the logarithm using a calculator, knowing \(\log\left(\frac{2}{3}\right) = \log 2 - \log 3\) reveals why the result is negative, specifically \(-0.1761\) in this case.
These properties not only help in manual calculations but also allow you to verify the results obtained via calculators by comparing them with the theoretical outcomes.

Decimal Approximation

When using a calculator to evaluate logarithms, you'll often see results expressed as a decimal approximation. This means the calculator provides a number that's very close to the true value of the logarithm, usually rounded to a specific number of decimal places.
For example, calculations for \(\log 2 \approx 0.3010\) or \(\log 35.2 \approx 1.5478\) are standardly rounded to four decimal places. This level of precision is typical for most practical uses, balancing accuracy and simplicity. When writing answers to problems, maintaining these four decimal places ensures that the response is both accurate and easy to interpret.
In the context of logarithms, such decimal approximations are significant because logarithmic values rarely resolve themselves into simple fractions or integers. Understanding this concept is vital, as it allows you to rely on approximations when exact calculations aren't feasible or necessary.