Problem 37 - Precalculus Mathematics for Calculus

Question

Use a calculator to evaluate the expression, correct to four decimal places. (a) \(\log 2\) (b) \(\log 35.2\) (c) \(\log \left(\frac{2}{3}\right)\)

Step-by-Step Solution

Answer

(a) 0.3010 (b) 1.5463 (c) -0.1761

Step 1: Understanding Logarithms

The logarithm of a number is the exponent to which the base (typically 10 for common logarithms) must be raised to produce that number. We will use a calculator with a log function to evaluate these expressions.

Step 2: Calculate \( \log 2 \)

Use a calculator to find \( \log 2 \). Enter 2 into your calculator and press the \( \log \) button. The calculator should display the result: 0.3010 (rounded to four decimal places).

Step 3: Calculate \( \log 35.2 \)

Enter 35.2 into your calculator and press the \( \log \) button. The result will be 1.5463, rounded to four decimal places.

Step 4: Calculate \( \log \left(\frac{2}{3}\right) \)

To find \( \log \left(\frac{2}{3}\right) \), first calculate the fraction \( \frac{2}{3} = 0.6667 \). Then enter 0.6667 into the calculator and press the \( \log \) button. The result should be -0.1761, rounded to four decimal places.

Key Concepts

Common Logarithms

Logarithms might sound complicated, but they’re actually quite straightforward when broken down. A common logarithm is a type of logarithm that uses 10 as its base. This is denoted by 'log', as opposed to \(' a \ ext {log} b \ ext {\} \)' which might use a different base. Essentially, when you see \(\log(x)\), it’s asking, “To what power does 10 need to be raised to give x?”

For instance:

• \( \log(100) \) = 2 because \( 10^2 = 100 \).
• \( \log(1000) \) = 3 because \( 10^3 = 1000 \).

Using this understanding, you can see that common logarithms are particularly useful in problems involving powers of ten, making calculations easier and more intuitive. They show up frequently in sciences and engineering fields.

Calculator Usage

Using a calculator to find logarithms is easy once you know the steps. Modern calculators have a specific button for common logarithms, typically labeled 'log'. This feature simplifies the process of finding logarithms without needing to manually do the math.

When using a calculator to evaluate a logarithm:

• Enter the number you want to find the logarithm of.
• Press the 'log' button.
• The calculator displays the result automatically.
• For precision, ensure that the calculator is set to give enough decimal points as required.

Let’s consider the example of calculating \( \log 35.2 \): Just enter ‘35.2’ and hit the 'log' key. The calculator will show the result, which you can round as needed. This method significantly saves time and reduces error that might occur in manual calculations.

Decimal Approximation

Calculating logarithms often results in non-whole numbers, which are usually expressed as decimal approximations. It’s essential to understand how to handle these approximations for clarity and accuracy.

When asked to approximate to a certain number of decimal places:

• Find the full decimal value from your calculator.
• Count the number of places after the decimal point.
• Round the number to the required precision (e.g., four places).

For example, if your calculator gives you a value of \( 1.546259 \) for \( \log 35.2 \), rounding it to four decimal places would give \( 1.5463 \). The process ensures precision in scientific calculations and helps communicate findings effectively.