Problem 49 - Algebra 2 and Trigonometry

Question

Solve each equation for the variable. \(\log _{2} 16+\log _{2} 2=\log _{2} x\)

Step-by-Step Solution

Answer

The value of \(x\) is 32.

Step 1: Simplify the Left Side

First, use the property of logarithms that states \(\log_b m + \log_b n = \log_b (mn)\). Since both logarithms on the left-hand side have the same base, 2, combine them: \(\log_{2} 16 + \log_{2} 2 = \log_{2} (16 \times 2)\).

Step 2: Evaluate the Product Inside the Logarithm

Calculate the product inside the logarithm: \(16 \times 2 = 32\). Therefore, the equation becomes \(\log_{2} 32 = \log_{2} x\).

Step 3: Solve for the Variable

Since the log expressions on both sides of the equation are equal and the base is the same, their arguments must also be equal. Thus, we can equate the values inside the logs without the logarithm: \(32 = x\).

Step 4: Conclusion

The value of \(x\) that satisfies the given equation is 32.

Key Concepts

Properties of Logarithms

Logarithms have unique properties that help simplify complex expressions. A key property, commonly used when dealing with logarithmic equations, is called the Product Property.

• The Product Property states: \[ \log_b (m) + \log_b (n) = \log_b (m \times n) \]

This means that when two logs with the same base are added, you can combine them into a single log by multiplying their arguments. For example:

• Given: \( \log_2 16 + \log_2 2 \)
• You can combine them: \( \log_2 (16 \times 2) \)
• Simplifying further: \( \log_2 32 \)

Understanding this property allows us to simplify expressions and solve logarithmic equations more easily.

Solving Equations

Solving logarithmic equations is a systematic process. Here's how we approached the exercise to solve for the variable \(x\):

• Firstly, use the properties of logarithms to simplify the equation, as we combined the logs on the left side using the Product Property.
• Next, calculate the combined product \(16 \times 2\) which equals 32, simplifying our logarithm to \(\log_2 32\).

The last step is crucial because when you have an equation like \(\log_b (m) = \log_b (n)\), you can deduce \(m = n\) since the base of the log on both sides is the same. Here, it translates the equation to:

• \(32 = x\)

This process illuminates how understanding log properties aids in finding solutions efficiently.

Logarithmic Functions

Logarithmic functions provide a means to unravel exponential equations. They are the inverse operations of exponential functions, therefore, they are crucial in many mathematical applications and real-world scenarios.A logarithmic function is generally written as \(y = \log_b x\), where \(b\) is the base, \(x\) is the argument, and \(y\) is the logarithmic value. The base \(b\) must be greater than 0 and different from 1.

• In our exercise, the log base is \(2\), making the function \(\log_2 x\).

Why is this important? Logarithmic functions help in:

• Solving equations where the power is unknown.
• Converting multiplicative processes into additive ones, simplifying operations.

These functions play an essential role both mathematically and practically, providing tools for simplifying calculations involving powers and roots. Understanding these concepts helps solve equations, like our example, and apply them in various problem-solving contexts.