Problem 30 - Algebra 2 and Trigonometry

Question

In \(24-35,\) for each given logarithm, find the antilogarithm, \(x .\) Write the answer to four decimal places. $$ \log x=4.8200 $$

Step-by-Step Solution

Answer

The antilogarithm of \( \log x = 4.8200 \) is approximately 66159.5000.

Step 1: Identify the given logarithmic equation

The problem states that \( \log x = 4.8200 \). To find the antilogarithm, we need to solve for \( x \).

Step 2: Apply the antilogarithm formula

The antilogarithm is the inverse operation of the logarithm. Thus, to find \( x \), calculate \( x = 10^{4.8200} \).

Step 3: Calculate the antilogarithm

Using a calculator, compute \( 10^{4.8200} \) which approximates to \( 66159.5000 \). Ensure the result is rounded off to four decimal places.

Key Concepts

Understanding Logarithms

Logarithms are a fascinating concept in mathematics that help us work with very large or very small numbers more conveniently. Essentially, a logarithm answers the question: "To what exponent must we raise a base number, usually 10, to achieve another number?" For example, if we have \( \log_{10} 100 = 2 \), this means 10 needs to be raised to the power of 2 to result in 100. Logarithms have a vital role in various fields such as science, engineering, and finance, where exponential growth or decay is involved.

• Base of logarithm: The number that is repeatedly multiplied in exponential form (commonly 10).
• Result: The power to which the base must be raised.

In the exercise given, we are working with \( \log x = 4.8200 \), meaning base 10 raised to what power equals \( x \)? The answer will uncover the antilogarithm, showing that understanding how to manipulate logarithms is key to solving real-world problems.

Inverse Operations and Antilogarithms

Inverse operations are mathematical processes that reverse each other’s effects. For instance, addition is the inverse of subtraction, while multiplication is the inverse of division. Similarly, logarithms and antilogarithms are inverse operations. If you know the logarithm of a number, you can find the original number by performing the antilogarithm operation—which is simply raising base 10 to the calculated power.

• Logarithm to antilogarithm: Switch via exponentiation.
• Antilogarithm calculation: Often involves powers of 10.

In the provided exercise, we want to discover the number \( x \) that, when plugged into the base 10 logarithm equation, gives us 4.8200. To find this, we compute \( 10^{4.8200} \), resulting in \( 66159.5000 \). By understanding these inverse relationships, solving logarithmic equations becomes a straightforward task.

Importance of Rounding Numbers

Rounding numbers is crucial when dealing with precise calculations, especially in scientific and engineering fields. It allows us to manage numbers consistently and communicate findings effectively without overloading with technically infeasible precision levels. For example, we often use rounding in statistics, mechanics, and computing to boost clarity and usefulness.

• Precision in calculation: Essential for accuracy.
• Communication: Simplifies complex data.

In our exercise, after performing the antilogarithm calculation \( 10^{4.8200} \), the result was \( 66159.5000 \). This value was rounded to four decimal places as instructed, allowing us to present a tidy and precise antilogarithm. Rounding ensures that we maintain a balance between simplicity and accuracy in calculations, pivotal in both academic work and real-life applications.