Problem 41 - Algebra 2

Question

Solve each equation. Check your answers. $$ \log (5-2 x)=0 $$

Step-by-Step Solution

Answer

The solution to the equation \( \log (5-2 x)=0 \) is \(x = 2\)

Step 1: Understand the equation

The exercise presents a logarithm equation. In logarithms, \(\log_a b=0\) is equivalent to \(a^0=b\), where the value of any number to the power of 0 is 1. Hence, \(\log_a b=0\) can be transformed into \(b = 1\). We have the equation as \(\log (5-2x)=0\), which can then be written as \(5 - 2x = 1\).

Step 2: Solve for x

Starting from \(5 - 2x = 1\), solving for x. Subtract 5 from both sides of the equation to get \(-2x = -4\). To solve for x, divide both side by -2, we get \(x = 2\) as the solutions.

Step 3: Check the result

To verify if this is the correct answer, substitute the solution, \(x = 2\), back into the original equation: \(\log (5-2x)=0 \rightarrow \log(5-2*2)=0 \rightarrow \log(5-4)=0 \rightarrow \log(1)=0\), which is true. Therefore, \(x = 2\) is the correct solution to the equation.

Key Concepts

Properties of Logarithms

Logarithms provide a unique way to handle exponential relationships. In its essence, a logarithm asks the question: "To what power must the base be raised to produce a given number?" In this exercise, the base is assumed to be 10 (the common logarithm), as it is not specified otherwise. Understanding the properties of logarithms is key when solving logarithmic equations.

One of these properties is that for any base "a", the equation \( \log_a (a^b) = b \) holds true. This property helps in simplifying logarithms back into normal equations by eliminating the log entirely.

Another useful property is when \( \log_a b = 0 \), which translates to \( b = a^0 \). In simpler terms, any number to the zero power equals 1. Hence in the equation \( \log (5-2x) = 0 \), the expression inside the log, "\(5-2x\)" can be simplified to equal 1.

Equation Solving Techniques

When we have transformed the logarithmic equation into a simpler form such as \( 5 - 2x = 1 \), we can use algebraic techniques to solve for the variable \( x \).

To isolate \( x \), follow these steps:

• First, subtract 5 from both sides to maintain balance of the equation. This gives \( -2x = -4 \).
• Next, solve for \( x \) by dividing both sides by -2. By doing so, the equation becomes \( x = 2 \).

Using these simple steps helps transform a complex looking logarithmic problem into a straightforward linear equation that is much easier to handle.

Algebraic Verification

Verification is an important step in solving any equation to ensure the solution is correct. This is especially crucial with logarithmic equations as there can be restrictions on the domain (i.e., what values \( x \) can take).

To verify, take the solution \( x = 2 \) and substitute it back into the original logarithmic equation: \( \log(5-2x) = 0 \).

By substituting \( x = 2 \), the expression becomes \( \log(5-4) = 0 \), which simplifies to \( \log(1) = 0 \).

Since \( \log(1) = 0 \) is a known true expression, the solution is confirmed to be correct. Always ensure the calculation checks out by plugging your solution into the original equation.