Problem 11 - Precalculus : Building Concepts and Connections

Question

In Exercises \(7-14,\) use \(\log 2 \approx 0.3010, \log 5 \approx 0.6990,\) and \(\log 7 \approx 0.8451\) to evaluate each logarithm without using a calculator. Then check your answer using a calculator. $$\log \sqrt{2}$$

Step-by-Step Solution

Answer

The value of \(\log \sqrt{2}\) is approximately 0.1505

Step 1: Transform the equation

First, we should transform \(\log \sqrt{2}\) into an easier form. We know that \(\sqrt{2}\) is \(2^{\frac{1}{2}}\), so we transform \(\log \sqrt{2}\) as \(\frac{1}{2} \log 2\).

Step 2: Substitute the value of \(\log 2\)

In the second step, substitute the given value of \(\log 2\) into our equation from step 1. Hence, we get \(\frac{1}{2} * 0.3010\).

Step 3: Calculation

In the last step, perform the multiplication operation from step 2. Doing the multiplication of \(\frac{1}{2} * 0.3010\) gets approximately 0.1505.

Key Concepts

Logarithm properties

Logarithms have some important properties that simplify mathematical calculations. These properties help in transforming complex expressions into simpler forms. Here, we will focus on the property used in this exercise, which is the power rule.

The power rule of logarithms states that if you have a logarithm of a number raised to a power, you can simplify it by multiplying the logarithm by that power. In mathematical terms:

• \(\log_b (x^n) = n \cdot \log_b x\)

In our exercise, we dealt with the expression \(\log \sqrt{2}\). Because \(\sqrt{2} = 2^{1/2}\), we can apply the power rule and write \(\log \sqrt{2}\) as \(\frac{1}{2} \cdot \log 2\). This transformation is what makes the logarithm easier to handle.

Logarithm calculation

Calculating logarithms can sometimes seem daunting without a calculator. But using known values and logarithm properties makes it manageable. In our specific exercise, after applying the power rule, we transformed the problem into a simple multiplication.

Given \(\log 2 \approx 0.3010\), the expression becomes \(\frac{1}{2} \cdot 0.3010\). To complete the calculation, we perform the multiplication:

• Multiply \(\frac{1}{2} \times 0.3010\)
• This equals approximately \(0.1505\)

By breaking down the calculation into these steps, finding the logarithm of \(\sqrt{2}\) becomes straightforward.

Mathematics problem solving

Solving mathematical problems like this one requires a combination of understanding and strategy. Start by analyzing the problem and identifying which mathematical rules or properties apply.

In this exercise, recognizing the need for the power rule enabled us to simplify the initial expression. The next step was to substitute the given value, \(\log 2 \approx 0.3010\).

• Transform complex expressions using known rules
• Substitute given values to simplify calculations
• Perform basic arithmetic to arrive at the solution

Finally, once you've done the manual calculations, it's beneficial to use a calculator to verify your result, ensuring both accuracy and a deeper understanding of the process. This way, mathematical problem solving both strengthens intuition and builds confidence.