Problem 91 - Intermediate Algebra

Question

Use a calculator to solve each equation. Round answers to four decimal places. See Example 6. $$ \log x=3.7813 $$

Step-by-Step Solution

Answer

\( x \approx 6037.0811 \)

Step 1: Recognize the Equation

The given equation is a logarithmic equation: \( \log x = 3.7813 \). This means we are looking for a number \( x \) such that the logarithm (base 10) of that number is 3.7813.

Step 2: Rewrite the Logarithmic Equation in Exponential Form

The equation \( \log x = 3.7813 \) can be rewritten using the definition of logarithms. Recall that if \( \log_{10} x = y \), then \( x = 10^y \). So we have \( x = 10^{3.7813} \).

Step 3: Use a Calculator to Compute

Using a calculator, compute \( 10^{3.7813} \). Enter the value 3.7813 into the calculator and use the exponent function to calculate the power of 10. The result is the value of \( x \).

Step 4: Round the Result

The calculator gives the result as approximately 6037.081055. Round this result to four decimal places, which provides \( x \approx 6037.0811 \).

Key Concepts

Understanding Logarithms

The concept of logarithms can sometimes seem complicated, but it boils down to something very simple. In essence, a logarithm answers the question, "To what power must we raise a base number to get another number?" In the common logarithm used here, the base is 10. For example, if \( \log_{10} x = 3.7813 \), we are looking for a number \( x \) such that when 10 is raised to a certain power (in this case, 3.7813), we get \( x \). This information gives us a powerful tool to solve equations involving exponentiation by translating them into a simpler form.

• A logarithm is essentially the inverse operation of exponentiation.
• Commonly used logarithms in mathematics are based on the numbers 10, e (natural logarithm), or 2 (binary logarithm).
• The notation \( \log x \) typically implies a base of 10, unless specified otherwise.

Grasping the idea of logarithms involves understanding their relationship with exponential forms, which leads us to the next topic.

Transforming Logarithmic Equations into Exponential Form

Converting a logarithmic equation into its exponential form is a vital step in solving the equation. The original logarithmic problem \( \log x = 3.7813 \) can be rewritten as an exponential equation because of the intrinsic relationship between logarithms and exponents. If you have \( \log_{b} x = y \), it means \( b^y = x \). Hence, for our exercise \( \log_{10} x = 3.7813 \), this means \( x = 10^{3.7813} \).

• The base, in this case, is 10, a standard choice for common logarithms.
• This transformation allows you to leverage the power of calculators to compute \( 10^{3.7813} \).
• An exponential form provides a direct computation path, which is often simpler than working with logarithmic expressions directly.

This conversion is a foundational skill in algebra and allows tackling a wide variety of problems that involve growth, decay, or amplifications.

Rounding Numbers

Rounding numbers is an essential mathematical skill, especially when dealing with measurements or calculations that result in long decimal expansions. Once we've calculated the exponential form, the calculator typically yields a precise result, such as 6037.081055. However, this precision is often more than we need, and we might be instructed to round the result for ease of use.

• To round a number to four decimal places, as the exercise suggests, observe the fifth decimal digit to decide if you round the fourth digit up or remain the same.
• In our example, the fifth decimal digit is 0, so the number remains at 6037.0811.
• Rounding helps in accurately presenting data in a manner that is easy to communicate and comprehend.

Practicing rounding off results not only saves space in our calculations but also aligns with precision needed for practical applications while ensuring the results are understandable and manageable.