Problem 98 - Elementary and Intermediate Algebra

Question

Use a calculator to solve each equation. Round answers to four decimal places. See Example \(6 .\) $$ \log x=1.4023 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 25.2867 \).

Step 1: Understanding the Logarithmic Equation

We are given the equation \( \log x = 1.4023 \). Here, the logarithm is in base 10, which is the common logarithm. We need to solve this equation for \( x \).

Step 2: Rewriting Logarithmic Form to Exponential Form

To isolate \( x \), we can rewrite the logarithmic equation in its exponential form. The equation \( \log x = 1.4023 \) can be converted to \( 10^{1.4023} = x \). This form identifies \( x \) as the result of 10 raised to the power of 1.4023.

Step 3: Using the Calculator

Now, use a calculator to compute \( 10^{1.4023} \). Most calculators have a button for computing powers of 10, often labeled as \( 10^x \). Enter 1.4023 and then press the \( 10^x \) button.

Step 4: Calculate and Round the Result

After computing \( 10^{1.4023} \), we find the result is approximately 25.2867. Rounding this to four decimal places, we have \( x = 25.2867 \).

Key Concepts

Logarithmic Equations

A logarithmic equation is an equation that involves the logarithm of an unknown variable. This type of equation can be particularly useful for finding the value of a variable when it is an exponent, making the problem easier to solve. In this original exercise, we began with the equation \( \log x = 1.4023 \). This equation suggests that "the common logarithm (or logarithm base 10) of \( x \) is 1.4023."

Logarithms are based on the concept of repeated multiplication. Therefore, a solution involves determining what power the base (in this case, 10) must be raised to, resulting in \( x \). Solving logarithmic equations often requires converting them into exponential form, which provides a clear path to finding the solution. Logarithmic equations can involve different bases, but when the base is 10, it is often referred to as a "common logarithm," simplifying calculations with typical calculators.

Overall, understanding logarithmic equations is an essential skill for solving many scientific and mathematical problems as they allow us to reverse the operation of exponentiation and find unknown exponent values.

Exponential Form

Exponential form is another way to represent a logarithmic equation. It expresses the same relationship, but instead of using a logarithm, it uses exponents. For instance, the logarithmic equation \( \log x = 1.4023 \) can be translated into exponential form as \( 10^{1.4023} = x \). This conversion helps in identifying what the value of \( x \) should be by calculating the result of raising the base (10) to the given power (1.4023).

Why do we convert to exponential form? It simplifies the calculation process. Instead of working with abstract logarithmic expressions, exponential form provides concrete arithmetic operations that are more straightforward to handle. This conversion not only clarifies the problem but also allows for easy computation using calculators.

Working with exponential equations is fundamental in various disciplines because it forms the backbone of exponential growth and decay models, such as those found in population dynamics, financial calculations, and radioactivity measurements.

Common Logarithm

The term "common logarithm" refers to logarithms with a base of 10. This base is "common" because it is the standard system used in everyday calculations, especially on scientific calculators which have dedicated functionalities for such computations. When we say \( \log x \), it is implicitly assumed to be \( \log_{10} x \).

Utilizing common logarithms can simplify the process of solving equations like \( \log x = 1.4023 \). The conversion to exponential form and subsequent calculation using base 10 offers a more intuitive approach, particularly given their prevalence in everyday devices and applications.

Common logarithms are widely used in science and engineering where measurements vary over vast ranges and powers of ten provide a convenient scale. Furthermore, they aid in performing operations on logarithmic scales, such as calculating pH in chemistry, sound intensity in decibels, and the Richter scale for earthquakes.

Overall, having a solid understanding of the common logarithm helps in effectively tackling various scientific and mathematical challenges that involve logarithms and exponential functions.