Problem 96 - Elementary and Intermediate Algebra

Question

Use a calculator to solve each equation. Round answers to four decimal places. See Example \(6 .\) $$ \log x=-0.926 $$

Step-by-Step Solution

Answer

The value of \( x \) is approximately 0.1187.

Step 1: Understanding the Problem

We are given an equation involving a logarithm: \( \log x = -0.926 \). The base of the logarithm is not specified, so we assume it is base 10, which is common for logarithms written as \( \log \). We need to solve for \( x \) by reverse-engineering the logarithm.

Step 2: Using the Properties of Logarithms

The logarithmic equation \( \log_{10} x = -0.926 \) can be rewritten in its exponential form as \( x = 10^{-0.926} \). This step relies on the property that \( \log_{b}(a) = c \) implies \( a = b^{c} \).

Step 3: Calculating the Exponential

Now, we need to compute \( 10^{-0.926} \) using a calculator. This involves raising 10 to the power of -0.926, which can be done using a calculator's power function (usually labeled \(^\) or \(x^{y}\)).

Step 4: Rounding the Result

After calculating \( 10^{-0.926} \) on the calculator, we get approximately 0.1187. Since the problem asks to round the answer to four decimal places, the value of \( x \) is 0.1187 after rounding.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation, involving two numbers, the base and the exponent. In our context, we deal with base 10, reflecting common logarithmic exercises. Exponentiation with base 10 transforms the number 10 raised to any given power. The process shows how many times we multiply 10 by itself. When dealing with negative exponents, such as in our exercise, the situation changes somewhat.

A negative exponent, like -0.926 in our example, indicates division rather than multiplication. Instead of multiplying, we divide 1 by the base raised to the absolute value of the exponent. For instance, when calculating \(10^{-0.926}\), think of it as \(\frac{1}{10^{0.926}}\). It flips our intuitive multiplication process into division.

When using a calculator, enter the base first, followed by the exponent. Ensure your calculator is set for handling negative and fractional exponents. It’s a simple operation but requires accuracy and understanding for correct results.

Properties of Logarithms

Logarithms simplify complex equations by transforming multiplication and division into addition and subtraction. They are inverse operations to exponentiation, bridging the gap between straightforward numerical operations and intricate calculation.

Consider the basic property: \(\log_{b}(a) = c\) means \(a = b^{c}\). This property allows us to convert logarithmic forms into exponential forms readily, as demonstrated in the problem where \(\log x = -0.926\) converts to \(x = 10^{-0.926}\). Recognizing this inverse relation is crucial for efficient problem-solving.

• Addition Property: \(\log_{b}(MN) = \log_{b}M + \log_{b}N\)
• Subtraction Property: \(\log_{b}\left(\frac{M}{N}\right) = \log_{b}M - \log_{b}N\)
• Power Property: \(\log_{b}(M^{n}) = n \cdot \log_{b}M\)

Memorizing these properties and applying them wisely is key to harnessing the full potential of logarithms in solving equations.

Rounding Decimals

Rounding decimals polishes our answers, making them more readable and practical for real-world applications. It’s the fine-tuning step after calculations, ensuring precision matches the problem’s requirements.

When rounding to four decimal places, focus on the fifth decimal digit. If it’s 5 or higher, increase the fourth decimal by one. If lower, retain the fourth as is. For example, if after calculation you find \(0.11865\), it rounds to \(0.1187\). This principle is simple yet requires careful attention to detail.

Common rounding rules:

• If the digit after desired place is 5 or more, round up.
• If less than 5, keep the digit unchanged.

Consistency in rounding ensures clarity and conforms to standardized answers found in educational or professional settings.