Problem 25 - Algebra 2 and Trigonometry

Question

In \(24-35,\) for each given logarithm, find the antilogarithm, \(x .\) Write the answer to four decimal places. $$ \log x=0.8297 $$

Step-by-Step Solution

Answer

The antilogarithm of \(0.8297\) is approximately \(6.7434\).

Step 1: Understanding the Problem

We need to find the antilogarithm of the given logarithm value 0.8297. This means finding the value of \(x\) such that \(\log_{10} x = 0.8297\).

Step 2: Definition and Formula

To find the antilogarithm of a logarithm, we use the exponential form. If \(\log_{10} x = y\), then the antilogarithm or \(x\) is given by \(x = 10^y\). So here, we must calculate \(10^{0.8297}\).

Step 3: Calculating the Antilogarithm

Using a calculator, compute the value of \(10^{0.8297}\). This involves raising 10 to the power of 0.8297, which can usually be done using scientific calculators or logarithm tables.

Step 4: Result and Rounding

Upon calculation, \(10^{0.8297} \approx 6.7434\). Ensure the result is rounded to four decimal places, as required by the problem statement.

Key Concepts

Logarithmic Functions

Logarithmic functions are crucial in mathematics for simplifying multiplication into addition, which can make complex calculations more manageable. A logarithm answers the question: _To what exponent must the base be raised, to produce a certain number?_ In simpler terms, it reverses the operation of an exponentiation.
For instance, the expression \( \log_{10} x = 0.8297 \) tells us that 10 needs to be raised to an unknown power to produce \( x \), and that power is 0.8297. Specifically, when the base is 10, it is called the common logarithm, often denoted simply as \( \log \).
Key points about logarithms to remember:

• The base can be any positive number, but common bases are 10, \( e \) (natural logarithm), and 2.
• Logarithms are used in various scientific calculations, including pH computations and decibel levels in acoustics.
• They have inverse relationships with exponential functions, which makes understanding exponential forms essential.

Exponential Form

The concept of exponential form is closely linked to logarithms. In an exponential form, a base number is raised to a certain power, which is expressed as \( y = b^x \). Finding the antilogarithm is essentially transforming a logarithmic statement back into this neat expression. This is why learning about exponentiation is important when dealing with logarithms.
In our case, since \( \log_{10} x = 0.8297 \), we express \( x \) in exponential form with: \( x = 10^{0.8297} \).
Let's explore the steps of using exponential form:

• Identify the base (often 10 for common logarithms).
• Use the inverse operation (exponentiation) with the logarithm value as the power.
• Calculate the result using a calculator for accuracy, especially when dealing with decimals.

Exponential forms are not just for logarithms, they appear in many areas of study from calculating interest in finance to growth rates in biology.

Rounding to Decimal Places

Rounding numbers is a fundamental skill, especially when dealing with decimals in scientific notation. After calculating the antilogarithm or any other precise value, the next crucial step is to round it according to the problem requirements.
For example, in the exercise given, we are asked to provide the answer to four decimal places. But why round decimals? It offers a simpler form while still providing a level of precision that is often sufficient.
Here's a quick guide to rounding:

• Identify the decimal place to which you need to round. The fourth decimal place is the number immediately after the third decimal point.
• Look at the number to the right of that place to decide whether to round up or down.
• If that number is 5 or greater, round up the last kept digit by 1. If less than 5, keep the last digit unchanged.

In our case, the calculated value from the antilogarithm was approximately 6.743429. Rounding to four decimal places gives us 6.7434, a practical and clear way to communicate the solution.