Problem 19 - Algebra for College Students

Question

Use your calculator to find \(x\) when given \(\log x\). Express answers to five significant digits. $$ \log x=-2.1928 $$

Step-by-Step Solution

Answer

\( x = 0.0063872 \)

Step 1: Understand the Problem

We need to find the value of \(x\) when \(\log x = -2.1928\). This means we are looking for a number whose logarithm to the base 10 is \(-2.1928\).

Step 2: Recall the Definition of Logarithms

The equation \(\log x = -2.1928\) can be rewritten in exponential form as \(x = 10^{-2.1928}\). This means \(x\) is the number when 10 is raised to the power of \(-2.1928\).

Step 3: Use a Calculator

Using a calculator, compute \(10^{-2.1928}\). Enter this expression into the calculator to find the value of \(x\).

Step 4: Calculate and Round Appropriately

Upon calculation, \(10^{-2.1928} \approx 0.0063872\). Since we need the answer to five significant digits, we round this result to \(0.0063872\).

Key Concepts

Base 10 Logarithms

The concept of base 10 logarithms is fundamental in understanding logarithmic equations. When we talk about the logarithm of a number, we are asking the question, "To what power do we need to raise 10 to obtain this number?" In mathematical terms, if \(\log_{10} x = a\), this means \(10^a = x\). It is crucial to note that the number 10 is the base here, which is standard in common logarithms, represented simply as \(\log x\) without any subscript.
Logarithms are particularly useful for solving equations involving exponential growth or decay. Once you understand that a logarithm is an exponent, it becomes easier to manipulate the equations. For the given exercise \(\log x = -2.1928\), imagining what power of 10 results in a tiny fraction helps us find that \(x\) quickly.
When solving problems involving logarithms, having a calculator with logarithmic function capability is beneficial. This allows you to switch between logarithmic and exponential forms efficiently.

Significant Figures

Significant figures are crucial when expressing and rounding the results of calculations. They represent the precision of a measured or calculated quantity. In this exercise, we are asked to express answers to five significant figures. Understanding what counts as a significant figure is key:

• All non-zero digits are considered significant.
• Any zeros between significant digits are also significant. For example, in "101," all digits are significant.
• Leading zeros are not significant. For example, in "0.0063872," these are only to locate the decimal point.
• Trailing zeros to the right of the decimal point are significant, reflecting precision. For example, "0.0063872" to five significant figures is still "0.006387" because it maintains precision.

Paying attention to significant figures ensures accuracy and reliability in quantitative data representation, which is vital in scientific and engineering calculations.

Exponential Form

Exponential form is a way of expressing numbers using a base and an exponent. It is particularly helpful in solving equations involving logarithms. To convert a logarithmic equation into exponential form, one can use the definition of logarithms: if \(\log_{10} x = a\), then \(x = 10^a\).
In this exercise, \(\log x = -2.1928\) translates directly to \(x = 10^{-2.1928}\). This conversion is crucial as it allows us to use a calculator to find the exact numerical value of \(x\). By recognizing that this exponential expression implies a very small positive number (since the exponent is negative), we can understand that the value of \(x\) will be much less than 1.
This form is effectively used in various fields including finance, physics, and computer science to represent very large or very small numbers in a more manageable format.