Problem 61 - Intermediate Algebra

Question

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ \log 2 x=\log 4 $$

Step-by-Step Solution

Answer

The exact solution is \( x = 2 \). An approximation is not needed as the solution is an integer.

Step 1: Use the Equality Property of Logarithms

Since logarithms have the property that if \( \log a = \log b \), then \( a = b \), we can equate the arguments of the logarithms. In our equation, this means \( 2x = 4 \).

Step 2: Solve for \( x \)

Solve the equation \( 2x = 4 \) by dividing both sides by 2 to isolate \( x \). This gives \( x = \frac{4}{2} = 2 \).

Key Concepts

Equality Property of Logarithms

Understanding the Equality Property of Logarithms is crucial for solving equations involving logarithms efficiently. This property tells us that if two logarithms with the same base are equal, then their arguments must also be equal. Essentially, when you see an equation like \( \log a = \log b \), you can conclude that \( a = b \), provided the base of the logarithms is the same. In practical terms, this means we can "drop" the logarithms from the equation entirely and just focus on the arguments. It simplifies the problem significantly, reducing the equation to one that is often much easier to handle using basic algebra.
In our original exercise, using the Equality Property allowed us to turn the problem \( \log 2x = \log 4 \) into the simpler equation \( 2x = 4 \). This crucial first step makes the rest of the solution much more straightforward. Overall, the Equality Property of Logarithms is a powerful tool in your mathematical toolkit, especially for solving logarithmic equations quickly and effectively.

Exact Solution

Finding the exact solution to an equation is about breaking it down to its simplest form where you can clearly determine the value of the variable in question. In our equation \( 2x = 4 \), after applying the Equality Property of Logarithms, we translate it to a simple algebraic operation.To solve for \( x \), we divide both sides by 2, isolating \( x \) on one side of the equation. This leads to:

• \( 2x = 4 \)
• \( \frac{2x}{2} = \frac{4}{2} \)
• \( x = 2 \)

These operations are simple enough, but it's important to understand them as part of the algebraic structure underlying the original logarithmic equation.
What makes this solution exact is that it doesn't rely on any approximations or rounding to a certain number of decimal places. It's straightforward, precise, and doesn't leave room for error or interpretation.

Approximations

In some cases, finding an approximation might be necessary or useful, especially when you're dealing with more complex equations or when instructed to furnish an answer with a specific number of decimal places. However, in our exercise, the calculation rendered \( x = 2 \), which is an integer and already in its exact form. There are no approximations needed here because the answer doesn't involve irrational or transcendental numbers that require rounding. However, understanding when and how to approximate is crucial. When faced with non-integer solutions, you will often need to round your answer.
Approximations are common in mathematics, science, and engineering:

• When the "exact" calculation is close but needs rounding for practical use.
• When using numerical methods to solve more complex equations or systems.
• When reporting measurements that have inherent uncertainty or precision limits.

Thus, while the original problem doesn't necessitate an approximate answer, being familiar with how and when to use approximations effectively can be invaluable in your study and application of mathematics.