Problem 92 - Elementary and Intermediate Algebra

Question

Use a calculator to solve each equation. Round answers to four decimal places. See Example \(6 .\) $$ \log x=2.8945 $$

Step-by-Step Solution

Answer

The solution to the equation is approximately 785.4139.

Step 1: Understand the equation

We are given the equation \( \log x = 2.8945 \). This means we have to find the value of \( x \) that satisfies this equation. \( \log x \) is in base 10 by default.

Step 2: Use the property of logarithms

To solve \( \log x = 2.8945 \), we use the property that \( x = 10^{\log x} \). So here, \( x = 10^{2.8945} \).

Step 3: Calculate using a calculator

Now, input \( 10^{2.8945} \) into your calculator. This will give you the value of \( x \).

Step 4: Round the result

Calculating \( 10^{2.8945} \) gives approximately 785.4138644. We need to round this to four decimal places, which results in 785.4139.

Key Concepts

Using Calculators in Algebra

Calculators are a crucial tool when tackling algebraic equations, especially those involving logarithms. They are designed to make computations faster and offer accurate results. When using a calculator for logarithmic problems, it's important to ensure that you are inputting the right function. Most calculators have a specific button for the logarithm base 10, usually labeled as "log."
It’s crucial to understand the syntax of your calculator:

• For calculating powers, you might use keys like "^" or buttons like "exp" depending on the model.
• After entering the base, input the exponent which in this case is provided in the equation (like "2.8945").

Using calculators not only simplifies calculations but also minimizes errors that come from manual computations, particularly in complex algebraic expressions. Remember to double-check the calculator’s settings to avoid any miscalculations, such as ensuring it’s in the correct mode (degree vs. radian) where applicable.

Rounding Numbers

Rounding is an essential mathematical skill that helps simplify numbers while maintaining their approximate value. In algebra, especially with logarithmic functions, rounding often comes into play to express the solution understandably. The process involves approximating a number to a specified number of decimal places or significant figures.
When rounding to four decimal places as shown in the solution, start by identifying the fourth decimal place. In the example, the number is 785.4138644.

• The fourth decimal place is "8," which we start with.
• Look at the next digit (which is "6" in the example). If it is 5 or higher, round the fourth place up by one. If it is lower than 5, leave it as it is.
• Thus, rounding 785.4138644 gives 785.4139.

Rounding ensures the result is more readable and practical. It’s also important in situations that require precision, especially in scientific and financial calculations. By practicing rounding, students gain confidence in simplifying complex numerical results.

Properties of Logarithms

Understanding the properties of logarithms is essential to solve logarithmic equations effectively. In the given problem, we used an important logarithmic property: if you know that \(\log_{b}x = y \), it follows that \( x = b^{y} \). In our specific exercise, the base is 10, leading to \( x = 10^{\log_{10}x} \).
Logarithms convert multiplicative processes into additive ones, making them easier to handle.

• Additional properties include \( \log_{b}(mn) = \log_{b}m + \log_{b}n \), showcasing how you can split the log of a product into the sum of logs.
• Another crucial property is \( \log_{b}\left(\frac{m}{n}\right) = \log_{b}m - \log_{b}n \).
• The power rule \( \log_{b}m^{n} = n\log_{b}m \) allows us to bring exponents in logs down to simplify the expression.

By mastering these properties, students can manipulate logarithmic expressions and solve complex equations more easily. Real-world applications include fields like biology, chemistry, and finance, where logarithmic scales measure things like pH, decibels, and compound interest.