Problem 93 - Elementary and Intermediate Algebra

Question

Use a calculator to solve each equation. Round answers to four decimal places. See Example \(6 .\) $$ \log x=-0.7630 $$

Step-by-Step Solution

Answer

\( x \approx 0.1738 \) after rounding to four decimal places.

Step 1: Understand Logarithm Basics

The equation given is \( \log x = -0.7630 \). In mathematics, when no base is specified, it is assumed to be base 10 (common logarithm). So, this equation implies \( \log_{10} x = -0.7630 \). This can be rewritten in exponential form.

Step 2: Convert Logarithmic Equation to Exponential Form

To solve the logarithmic equation \( \log_{10} x = -0.7630 \), convert it to its exponential form: \[ x = 10^{-0.7630} \] This conversion is based on the definition of a logarithm: that \( a^b = c \) is equivalent to \( \log_a c = b \).

Step 3: Use a Calculator to Compute Value

Input the expression \( 10^{-0.7630} \) into a calculator to find the value of \( x \).Ensure the calculator is in the correct mode for common logarithms (base 10). The result will be the value of \( x \).

Step 4: Round to Four Decimal Places

The calculator gives a preliminary result, which must be rounded to four decimal places as the problem requires. Using decimal rounding rules, ensure that only four digits are retained after the decimal point for the answer.

Key Concepts

Exponential Form

Understanding the concept of exponential form can simplify solving logarithms and reveal the relationship between them. When tackling a logarithmic equation like \( \log_{10} x = -0.7630 \), converting it to exponential form can be crucial. This conversion is based on the principle that the logarithm represents an exponent.Here’s how it works:

• The equation \( \log_{10} x = -0.7630 \) implies the power to which 10 must be raised to yield \( x \) is \(-0.7630 \).
• In exponential form, this translates to \( x = 10^{-0.7630} \).

This step is not only about finding an expression for \( x \) but understanding sections of exponential and logarithmic expressions that mirror each other. It essentially bridges the gap between the two concepts, allowing us to solve for \( x \) by computing the power of 10 that yields our answer.

Common Logarithm

When you see \( \log x \) without any specific base attached, it often refers to the common logarithm. The common logarithm is logarithm base 10, which is widely used in various mathematical calculations and scientific work due to its adaptability with decimal systems.

• This notation simplifies to \( \log_{10} x \).
• Common logarithms are highly prevalent in equations when working with exponential scales like sound intensity or pH levels.

By understanding that "common logarithm" has an assumed base of 10, we can deal more comfortably with equations, properly applying the right methods in calculations. This standardization makes it easier to switch between different forms of expressions, allowing streamlined calculation approaches.

Calculator Usage

Using a calculator effectively for logarithmic and exponential problems is key, especially when converting forms or finding numerical values. Calculators are designed to handle both logarithmic and exponential calculations, ensuring precision.Here’s how to approach using a calculator for our example:

• Ensure your calculator settings are correct, particularly set for base 10 since we’re dealing with the common logarithm.
• Input the exponential form directly: \( 10^{-0.7630} \).
• Most scientific calculators offer a button for powers of 10, often denoted as \( \text{10}^x \) or simply \( ^a \).

With correct usage, the calculator makes it easy to evaluate even complex prescriptions quickly. Proper understanding and familiarity with your calculator’s functions will not only make solving such exercises more efficient but also more accurate.

Decimal Places

Exactness in mathematical results often requires that answers comply with the specified number of decimal places; precision and accuracy are vital.To satisfy the problem requirement to round to four decimal places:

• Once the calculator has provided the solution, identify the first four numbers after the decimal point in the result.
• Use standard rounding rules to determine if the fourth number needs adjustment based on the fifth number. If it’s 5 or greater, the fourth number goes up by one.

For example, if the calculator shows \( x = 0.173784 \, \), adjusting for four decimal places results in \( 0.1738 \). Paying attention to these details helps maintain precision, ensuring your answers are as close to the exact value as possible.