Problem 14 - Algebra for College Students

Question

Use your calculator to find \(x\) when given \(\log x\). Express answers to five significant digits. $$ \log x=3.9335 $$

Step-by-Step Solution

Answer

x is approximately 8569.3.

Step 1: Understanding the Problem

We are given \( \log x = 3.9335 \) and are asked to find the value of \( x \). The logarithm here is common, i.e., base 10, which means \( \log_{10} x = 3.9335 \).

Step 2: Exponential Form Conversion

To find \( x \), we need to convert the logarithmic expression to its exponential form. This can be done by using the principle that if \( \log_{10} x = y \), then \( x = 10^y \). So, \( x = 10^{3.9335} \).

Step 3: Using the Calculator

Enter \( 10^{3.9335} \) into the calculator. This will give us the value of \( x \). Check that the calculator is set to display five significant digits.

Step 4: Calculation Result

Upon calculation, \( 10^{3.9335} \approx 8569.3 \). Therefore, \( x \approx 8569.3 \) to five significant digits.

Key Concepts

Exponential Form

In mathematics, converting a logarithmic expression into its exponential form is a fundamental skill. It allows us to simplify the process of finding unknown values. When you see an equation like \( \log_{10} x = y \), you can convert it to exponential form to solve for \( x \). This transformation follows a basic rule:

• If \( \log_{10} x = y \), then \( x = 10^y \).

This means that the expression \( \log_{10} x = 3.9335 \) can be rewritten as \( x = 10^{3.9335} \) in exponential form. This step is crucial as it allows us to unlock the power of exponentiation to find the value of \( x \).
Exponential form is often simpler for calculations because it provides a direct way to evaluate the expression using a calculator. By understanding this, students can tackle logarithmic problems with ease and precision.

Common Logarithm

The term "common logarithm" refers to a logarithm with a base of 10. This is the most frequently used base due to its direct correlation with our decimal number system. When a logarithm is expressed without a base, as in \( \log x \), it is typically understood to be a common logarithm.

• Common logarithms are useful in a variety of fields, including science and engineering.
• They help in transforming multiplicative relationships into additive ones.

Understanding common logarithms is fundamental as they simplify complex calculations, making it easier to work with large numbers. In our exercise, the notation \( \log x = 3.9335 \) directly implies a base of 10, which tells us to use common logarithm principles for the solution.

Significant Digits

Significant digits (or figures) in a numerical value convey the precision of that value. It indicates the confidence in the digits' accuracy. When performing calculations, especially with logarithms, it is important to round the answer to the required number of significant digits. This ensures consistency and precision in results.

• In this problem, we are instructed to express the result to five significant digits.
• Such precision is necessary in scientific calculations where exact measurements matter.

For example, after calculating \( 10^{3.9335} \), you obtain a rough value of 8569.3. Each digit in this number plays a role in maintaining the integrity of the result within the specified significant digits. Keeping track of significant figures adds clarity and accuracy to mathematical solutions.

Calculator Usage

Using a calculator efficiently is a vital skill when dealing with logarithms and exponential expressions. Calculators save time and improve accuracy if used correctly. Here are some tips to consider when using a calculator:

• Ensure the calculator is set to display results in the correct number of significant digits.
• Familiarize yourself with the scientific notation and exponential functions on your calculator.

For this exercise, inputting \( 10^{3.9335} \) into your calculator will compute \( x \). Confirm your calculator's settings allow for smooth calculation without cutting off significant digits.
By practicing with calculators, students can build confidence to handle a variety of computations, making complex math problems more approachable.