Problem 16 - Intermediate Algebra

Question

For Problems \(11-20\), use your calculator to find \(x\) when given \(\log x\). Express answers to five significant digits. $$ \log x=0.5517 $$

Step-by-Step Solution

Answer

\( x \approx 3.5704 \)

Step 1: Understanding the Problem

We are given the logarithmic equation \( \log x = 0.5517 \). Our task is to find the value of \( x \) that satisfies this equation.

Step 2: Rewriting the Logarithmic Equation

Recall that if \( \log x = a \), then \( x = 10^a \). Therefore, our equation \( \log x = 0.5517 \) can be rewritten as \( x = 10^{0.5517} \).

Step 3: Calculating the Antilogarithm

Use a calculator to find \( x = 10^{0.5517} \). This calculation will give us the numerical value of \( x \).

Step 4: Rounding the Answer

Once you calculate \( x \), round the answer to five significant digits. This step ensures that the answer meets the required precision.

Step 5: Final Answer

Based on calculations, the value of \( x \) to five significant digits is approximately 3.5704.

Key Concepts

Understanding Antilogarithm

In mathematics, an antilogarithm is the inverse function of a logarithm. If you are given a logarithmic value and want to find the original number before the logarithm was applied, you need to calculate the antilogarithm. This process involves exponentiation. For any equation of the form \( \log x = a \), the antilogarithm helps to find \( x \), which can be rewritten as \( x = 10^a \).

Essentially, you are using base 10 as the term for exponential power. Suppose \( \log x = 0.5517 \), then \( x \) becomes \( 10^{0.5517} \). Calculating antilogarithms is a crucial step in solving real-world problems where logarithmic scales are used.

Using a calculator to directly find the antilogarithm by exponentiating the given logarithmic number is often faster and easier. Remember, antilogarithms are fundamental when working with any data that involves exponential growth, such as population growth, radioactive decay, or even calculations in financial sectors.

The Importance of Significant Digits

Significant digits (or significant figures) are the digits in a number that contribute to its precision. When you solve mathematical problems, you must present your answer with the correct number of significant digits, as this reflects the certainty and precision of your calculation.

To illustrate, if you're given a different problem where you are instructed to express your result to three significant digits, your calculation and final result must reflect this standard. In this particular exercise, you need to express the value of \( x \) with five significant digits. Therefore, if the calculator provides a value like 3.570416, you would round it to 3.5704 to keep the answer concise to five significant figures.

• All non-zero digits are always significant.
• Any zeros between significant digits are significant.
• Leading zeros are not significant.
• Trailing zeros in a decimal number are significant.

By understanding significant digits, you ensure that your calculated answers are accurate and precise, which is particularly important in scientific and engineering contexts.

Calculator Usage Tips

Calculators are powerful tools in solving mathematical problems, especially when it involves calculating antilogarithms. To efficiently use your calculator for such problems, follow these simple steps:

Firstly, ensure your calculator is in the correct mode. For logarithmic calculations like finding antilogarithms, the scientific mode is often required. Most scientific calculators have a designated button for exponentiation or calculating antilogs labeled as '10^x' or something similar.

When you enter a logarithmic value such as 0.5517, locate the antilog or exponential function button and key in the full number. By pressing the right sequence, the calculator will compute \( 10^{0.5517} \) and provide the value of \( x \).

Once you've found the value of \( x \), make sure to double-check the result by reviewing that the calculator's screen isn't showing too many or too few digits. Practicing this makes rounding to significant digits easier.

Using calculators correctly not only speeds up problem-solving but also aids in maintaining accuracy and consistency in mathematics and beyond.