Problem 94 - Elementary and Intermediate Algebra

Question

Use a calculator to solve each equation. Round answers to four decimal places. See Example \(6 .\) $$ \log x=-1.3587 $$

Step-by-Step Solution

Answer

The solution to \( \log x = -1.3587 \) is approximately \( x = 0.0438 \).

Step 1: Understand the Equation

The equation given is \( \log x = -1.3587 \). The goal is to find the value of \( x \). The logarithm here is assumed to be base 10 (common logarithm).

Step 2: Apply the Inverse Operation

To solve for \( x \), we need to rewrite the logarithmic equation in its exponential form. The inverse of a base-10 logarithm is the exponentiation to base 10: \( x = 10^{-1.3587} \).

Step 3: Calculate the Exponential Value

Using a calculator, compute \( 10^{-1.3587} \). Enter the value \( -1.3587 \) into the calculator and find the power of 10 raised to this value.

Step 4: Round Your Answer

The result from the calculator will provide a decimal number. Round this number to four decimal places as the problem requests.

Key Concepts

Exponential Form

To solve equations involving logarithms, it's crucial to understand "exponential form." The exponential form is a way to express relationships in terms of a base raised to an exponent. For example, if we have a logarithm such as \( \log_{10} x = y \), it means that \( 10^y = x \). This conversion is based on the property that the logarithm is the inverse of exponential functions, specifically descriptive of the power that a base number is raised to in order to obtain another number. This form simplifies the process as it allows us to directly calculate the value of \( x \), which can be powered by using a calculator. Understanding this conversion—and how it applies to shifting from logarithmic to exponential form—is a fundamental technique for solving logarithmic equations effectively.
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The generic form is \( b^y = x \), where "b" is the base of the logarithm, and "y" is the logarithm value.

This form helps break down complex logarithmic expressions into solvable exponentials.

Base 10 Logarithm

The base 10 logarithm, often referred to as the "common logarithm," is an extremely practical tool in mathematics. It expresses how many times you would multiply the number 10 to reach another number. This is represented by \( \log_{10} x \). Consider this the fundamental base of most calculations involving common logarithms. The assumption in mathematical problems is often that if no base is provided alongside the \( \log \) notation, it defaults to base 10. Let's break down why this is helpful:

• Base 10 is simple and corresponds to our decimal numbering system, making it intuitive.
• Scientific and engineering fields frequently use base 10 for its ease of computation and widespread calculator support.
• The operations on base 10 allow straightforward transition into exponential forms, using a standard conversion process.

Thus, understanding the base 10 logarithm gives a strong foundation for solving complicated equations and examining exponential growth or decay.

Rounding Decimals

When it comes to real-world calculations, precision often matters, and that's where "rounding decimals" takes the stage. After using a calculator to find a number from an exponent operation like \( 10^{-1.3587} \), you will usually end up with a long decimal. The exercise then asks us to round the result to four decimal places. Here’s what you need to know about rounding to four decimal places:

• Look at the fifth decimal digit. If it is 5 or higher, round the fourth decimal place up by one.
• If the fifth digit is less than 5, leave the fourth decimal place as it is.
• This rule helps ensure accuracy and usability of numbers within realistic contexts.

Rounding makes numbers simpler to work with, while still close to the original value. Thus, understanding rounding is key to solving many mathematics problems that call for approximations.

Inverse Operations

Inverse operations undo a function's operation. Understanding inverse operations is crucial in mathematics, especially for solving equations. For logarithms, exponentiation serves as their inverse operation. Here’s how it applies:

• The log function tells you which power to raise a base to obtain a number.
• Exponentiation then uses this power to return to the number from the logarithm output.
• In the given exercise, moving from \( \log x = -1.3587 \) to \( x = 10^{-1.3587} \) demonstrates this concept practically.

This relationship showcases the beauty of mathematics, as it provides a reliable way to navigate complex equations by simplifying them through inverse processes. Mastering inverse operations is a necessary skill for anyone looking to adeptly tackle mathematical problems.