Problem 50 - Algebra 2 and Trigonometry

Question

Solve each equation for the variable. \(\log _{2} x-\log _{2} 8=\log _{2} 4\)

Step-by-Step Solution

Answer

The solution is \( x = 32 \).

Step 1: Convert Subtraction to Division

We have the equation \( \log_{2} x - \log_{2} 8 = \log_{2} 4 \). We can use the logarithmic identity \( \log_{b} A - \log_{b} B = \log_{b} \left( \frac{A}{B} \right) \) to combine the logs on the left-hand side: \( \log_{2} \left( \frac{x}{8} \right) = \log_{2} 4 \).

Step 2: Eliminate Logarithms

Since the log on both sides of the equation have the same base, we can set the arguments equal to each other: \( \frac{x}{8} = 4 \).

Step 3: Solve for x

To solve for \( x \), multiply both sides of the equation by 8 to get rid of the fraction: \( x = 4 imes 8 \). This simplifies to \( x = 32 \).

Key Concepts

Logarithmic Identity

The concept of logarithmic identity is crucial for solving equations involving logarithms. This identity, particularly important, lets us simplify expressions involving the logarithm of a quotient.
The rule states:

• If you have a subtraction of two logarithms with the same base, such as \( \log_{b} A - \log_{b} B \), it is equivalent to \( \log_{b} \left( \frac{A}{B} \right) \).

This identity is powerful because it allows us to condense two logarithms into one. In the example equation \( \log_{2} x - \log_{2} 8 = \log_{2} 4 \), we apply this identity to combine the logarithms on the left side:

• The left side becomes \( \log_{2} \left( \frac{x}{8} \right) \).

This simplification is the first step towards isolating the variable and solving the equation.

Solving Equations

Once your logarithmic terms are combined using the logarithmic identity, the next step is solving the equation. Many logarithmic equations can be simplified in such a way that allows removing the logarithmic terms entirely.
In the example equation, after applying the identity, both sides of the equation become equal logarithms.

• The equation becomes \( \log_{2} \left( \frac{x}{8} \right) = \log_{2} 4 \).

To solve the equation, it’s key to recognize that if \( \log_{b} \left( C \right) = \log_{b} \left( D \right) \), then \( C = D \). This insight helps us skip the log entirely and set the inner parts equal:

• This results in \( \frac{x}{8} = 4 \).

This equation is simpler to solve, focusing on isolating the variable \( x \).
This technique of setting the contents of the logarithms equal to each other is a valuable tool when dealing with logarithmic equations.

Base of Logarithms

The base of logarithms plays a pivotal role in understanding logarithmic functions and solving logarithmic equations. In any logarithmic function like \( \log_{b} x \), \( b \) is the base. It determines the behavior and scaling of the logarithmic curve.
Each base makes the function unique, whether it’s base 2, base 10, or another number.
When solving equations, having consistent bases on both sides of the equation is necessary:

• This consistency ensures that operations like combining logs or comparing their contents (as we did in setting \( \log_{2} \left( \frac{x}{8} \right) = \log_{2} 4 \)) are valid.
• If the bases are inconsistent, we can’t directly apply these operations.

For practical solving, always check the base. Often, exercises and real-world problems use common bases like 2, 10, or \( e \), but knowing how to handle different bases expands your problem-solving toolkit. Understanding the base can also help with real-life scenarios where logarithms apply, such as measuring sound intensity or earthquake magnitudes.