Problem 18 - Algebra for College Students

Question

Use your calculator to find \(x\) when given \(\log x\). Express answers to five significant digits. $$ \log x=-0.1452 $$

Step-by-Step Solution

Answer

\( x = 0.71508 \)

Step 1: Understand the logarithmic equation

The problem provides us with a logarithmic equation: \( \log x = -0.1452 \). Here, \( \log x \) refers to the common logarithm, which is the logarithm with base 10. We need to find the value of \( x \) such that its base 10 logarithm equals -0.1452.

Step 2: Rewrite the equation in exponential form

Since \( \log_{10} x = -0.1452 \), we can rewrite this in exponential form using the definition of a logarithm: \( x = 10^{-0.1452} \). This means \( x \) is the result of raising 10 to the power of -0.1452.

Step 3: Use a calculator to compute the value of \( x \)

Enter the expression \( 10^{-0.1452} \) into a calculator to determine the value of \( x \). Ensure the calculator is set to a precision that provides results to at least five significant digits.

Step 4: Record the result

After calculating \( 10^{-0.1452} \) with a calculator, the result is approximately \( 0.71508 \). Make sure to express this result to five significant digits as instructed.

Key Concepts

Exponential Form

When dealing with logarithmic equations, one powerful method is converting them into exponential form. This conversion helps to simplify the problem and solve for the unknown in a more direct way.
The idea is based on the property that if you have a logarithmic equation \( \log_{b}(a) = c \), then this can be rewritten in its equivalent exponential form as \( b^{c} = a \).
In this context, the base \( b \) is often 10, especially when we talk about common logarithms. Here, it helps to understand what each part of the equation represents:

• \( b \): The base of the logarithm, which is 10 for common logarithms.
• \( c \): The exponent that the base is raised to in order to get \( a \).

Understanding these transformations between logarithmic and exponential forms is crucial because it can greatly simplify problem-solving and calculating exact values when using a calculator.

Logarithmic Equations

Logarithmic equations are a type of equation that include logarithms. These equations express the concept of exponentials in a different way. The common logarithm, which typically involves base 10, is used frequently in various mathematical applications.
Learning to understand and solve logarithmic equations involves recognizing their relationship with exponential forms.
This is because logarithms effectively "undo" exponentiation, and vice versa:

• For example, \( \log_{10}(x) = y \) means 10 raised to the power of \( y \) gives you \( x \).

To solve these equations, it's important to know:

• How to isolate the logarithmic part to one side of the equation.
• The process of converting to exponential form to solve for the unknown variable.
• Using properties of logarithms to simplify calculations.

Practicing solving these equations can significantly improve your mathematical skills and understanding of exponential relationships.

Precision in Calculations

Precision in calculations is a key aspect to consider, especially when deploying calculators or digital tools in finding solutions.
Ensuring precision and accuracy is important to achieve reliable results, particularly when your answer must be to a specified number of significant digits.
When performing a calculation such as \( 10^{-0.1452} \), several steps must be taken to maintain precision:

• Make sure the calculator is set up to display enough digits. Here, at least five significant digits are specified.
• Understand that significant digits are the meaningful digits in a number, which inform you of the precision of the measurement or calculation.
• Be meticulous in entering values into the calculator, avoiding rounding errors prematurely.

Applying these principles when using calculators ensures your results are reliable and accurate, especially in fields requiring high precision, such as engineering or physics.