Problem 20 - Intermediate Algebra

Question

For Problems \(11-20\), use your calculator to find \(x\) when given \(\log x\). Express answers to five significant digits. $$ \log x=-2.6542 $$

Step-by-Step Solution

Answer

\( x = 0.0022122 \)

Step 1: Identify the Given Equation

We are provided with the equation \( \log x = -2.6542 \). Our task is to solve for \( x \).

Step 2: Convert the Logarithmic Equation to Exponential Form

Recall that if \( \log_{10}(x) = y \), then \( x = 10^y \). Therefore, convert the equation \( \log x = -2.6542 \) to its exponential form, giving us \( x = 10^{-2.6542} \).

Step 3: Calculate the Exponential Value

Next, use a calculator to evaluate \( 10^{-2.6542} \). Enter this expression into the calculator to compute the value of \( x \).

Step 4: Record the Result with Correct Precision

The result calculated is approximately \( x = 0.0022122 \). Make sure to record it with five significant digits as the problem instructs.

Key Concepts

Exponential Form Conversion

Understanding how to convert a logarithmic equation into its exponential form is a key skill when solving logarithmic equations like \( \log x = -2.6542 \). This process involves recognizing the relationship between logarithms and exponents.

In general, if you have \( \log_{10}(x) = y \), you can convert this to an exponential equation by expressing \( x \) as \( 10^y \).
This makes use of the fact that a logarithm answers the question: "To what power must the base (10, in this case) be raised, to produce a given number?"

By applying this conversion to our problem, we changed the logarithmic equation \( \log x = -2.6542 \) into its exponential form: \( x = 10^{-2.6542} \).

This form is much more straightforward to compute and resolve using a calculator tool. Always ensure you understand this conversion process as it's fundamental to solving logarithmic equations.

Using Calculators in Math

When solving equations involving complex numbers or precise decimals, a calculator becomes an invaluable tool. The equation \( x = 10^{-2.6542} \) is an example where the calculation is simplified significantly using a calculator.

Here’s how you can approach using a calculator for such problems:

• Make sure your calculator is set to the correct mode (usually scientific mode) to handle exponential calculations properly.
• Enter the base (10 in this case) followed by the exponent (-2.6542).
• Use the power or exponentiation button (often labeled as \( 10^x \) or similar) to find the result.
• Verify that no unwarranted rounding occurs by checking the settings for decimal precision.

By mastering the use of a calculator, you enhance your efficiency and ensure accuracy whenever dealing with exponential calculations.

Precision in Mathematical Calculations

Precision is critical when performing mathematical computations, especially those involving logarithms and exponents. For exercises requesting results up to a specific number of significant digits, like five in this exercise, it’s crucial to handle rounding correctly to maintain accuracy.

Significant digits are the digits that carry meaning contributing to a number's precision. In our calculation, we found \( x = 0.0022122 \).

To express this answer to five significant digits, confirm that your calculator is set up to display enough digits or manually round off your result.

• Round off any non-significant digits only after you've ensured the correct precision.
• Always cross-check the final figure to avoid errors in precision that could affect the outcome significantly.

Understanding these concepts ensures your calculations are both precise and reliable.