Problem 9 - Algebra 2 and Trigonometry

Question

Solve each equation for the variable and check. \(\log 8-\log x=\log 2\)

Step-by-Step Solution

Answer

The solution for \( x \) is 4.

Step 1: Simplify Logarithmic Equation

The given equation is \( \log 8 - \log x = \log 2 \). Using the property of logarithms that states \( \log a - \log b = \log \frac{a}{b} \), we can rewrite the equation as: \[ \log \frac{8}{x} = \log 2 \].

Step 2: Equate Arguments

Since the logarithms are equal, their arguments must also be equal. Therefore, we have: \[ \frac{8}{x} = 2 \].

Step 3: Solve for x

To isolate \( x \), multiply both sides of the equation by \( x \) to get \( 8 = 2x \). Then, divide both sides by 2 to solve for \( x \): \[ x = \frac{8}{2} = 4 \].

Step 4: Verification

Substitute \( x = 4 \) back into the original equation to verify the solution: \( \log 8 - \log 4 = \log 2 \). Using the property \( \log a - \log b = \log \frac{a}{b} \), this simplifies to \( \log \frac{8}{4} = \log 2 \). Since \( \frac{8}{4} = 2 \), it checks out as \( \log 2 = \log 2 \). Hence, the solution is verified.

Key Concepts

Properties of Logarithms

Logarithms can be tricky, but understanding their properties makes solving logarithmic equations much easier. One significant property of logarithms is the subtraction rule:

• When you subtract two logarithms with the same base, such as \(\log a - \log b\), you can combine them into a single logarithm: \(\log \frac{a}{b}\).
• This means you're effectively dividing the arguments of the logarithms.

This property is handy when you want to simplify expressions or equations involving multiple logarithms. It helps convert complicated logs into simpler forms.
In the example, we have \(\log 8 - \log x = \log 2\). Applying the subtraction property, we combine the logs into: \(\log \frac{8}{x} = \log 2\).
This transformation sets the stage to compare and verify the arguments on both sides of the equation.

Verification of Solutions

After finding a solution to a logarithmic equation, it's crucial to verify that the solution is correct. Verification confirms that the steps taken during the solving process maintain mathematical accuracy.
Once you solve for a variable, substitute the solution back into the original equation. As demonstrated in our current task:

• Original equation: \(\log 8 - \log x = \log 2\)
• Substitute \(x = 4\) into the equation
• The expression becomes \(\log 8 - \log 4 = \log 2\)

Using the properties of logarithms tells us that this simplifies to \(\log \frac{8}{4} = \log 2\). Since \(\frac{8}{4}\) equals 2, our simplified equation is rightly \(\log 2 = \log 2\), verifying our result.
This step assures us that not only is the solution calculated correctly, but it also fulfills the original equation, ensuring no oversights occurred.

Solving Equations

Solving equations with logarithms requires a keen understanding of applying the properties of logarithms and algebraic manipulation. Here is a methodical approach:
Start by simplifying the equation. For the equation \(\log 8 - \log x = \log 2\), simplify it using the properties of logarithms. Turn it into \(\log \frac{8}{x} = \log 2\).
The goal is to isolate the variable. Recognize that if two logarithms with the same base are equal, their arguments must also be equal. Thus for our equation: \(\frac{8}{x} = 2\).

• To solve for \(x\), multiply both sides by \(x\): \(8 = 2x\).
• Next, isolate \(x\) by dividing each side by 2: \(x = \frac{8}{2}\).
• Finally, \(x\) simplifies to 4.

Each step is a logical progression that follows algebraic rules, leading to a clear solution. This method ensures accuracy and efficiency when dealing with logarithmic equations.