Problem 48 - Algebra A Combined Function

Question

Solve. $$ \log _{2} x=3 $$

Step-by-Step Solution

Answer

The solution is \( x = 8 \).

Step 1: Understanding the Logarithmic Equation

The equation given is \( \log_{2} x = 3 \). This logarithmic equation can be understood as asking the question: "To what power must 2 be raised, to result in \( x \)?"

Step 2: Convert to Exponential Form

From the understanding of logarithms, we know that the equation \( \log_{b} a = c \) can be rewritten as an exponential equation: \( b^{c} = a \). Therefore, \( \log_{2} x = 3 \) converts to the exponential equation \( 2^{3} = x \).

Step 3: Calculate the Exponentiation

Now, solve the exponential equation \( 2^{3} = x \). Calculate \( 2^{3} \) by multiplying 2 by itself three times: \( 2 \times 2 \times 2 = 8 \).

Step 4: Solution

After solving the exponential equation, we find that \( x = 8 \). Thus, the solution to the equation \( \log _{2} x = 3 \) is that \( x = 8 \).

Key Concepts

Exponential Form

The exponential form is a crucial concept in mathematics used to simplify expressions. When converting a logarithmic equation to an exponential form, you rewrite the equation to express how many times a base number must be multiplied by itself to reach a certain value. In the given example, the logarithm is \( \log_{2} x = 3 \), which asks the question "To what power must 2 be raised to get \( x \)?"

By converting this into exponential form, we write it as \( 2^{3} = x \). Here, 2 is the base, and 3 is the exponent, indicating that 2 must be raised to the power of 3. This conversion helps us see that the solution involves simple exponentiation.

Understanding how to transition from a logarithmic equation to an exponential form is vital in solving equations efficiently. It uses the fundamental concept that logarithms are the inverse of exponentiation.

Solving Equations

Solving equations is about finding the value of the unknown variable that makes the equation true. The original problem \( \log_{2} x = 3 \) requires both algebraic manipulation and number sense to solve.

Once the equation is converted into an exponential form \( 2^3 = x \), solving it becomes straightforward. Here, you compute the right side: calculating \( 2 \times 2 \times 2 \). This results in 8, hence \( x = 8 \).

This step-by-step approach ensures clarity and accuracy. Each step simplifies the problem until you reach the solution. Always check your calculations to make sure they align with the logic of the operation being performed.

Logarithmic Functions

Logarithmic functions are an essential tool in algebra that help simplify exponential relationships. They are simply the inverse operation of exponentiation. When dealing with equations involving logarithms, understanding how these functions work can make problem-solving more accessible.

In the context of the example, \( \log_{2} x = 3 \) involves a log function with base 2. The logarithm represents the power to which we need to raise the base (2) to achieve the number \( x \). Knowing this, we can transform it easily into exponential form.

Mastering logarithmic functions involves understanding their properties, such as:

• The product rule: \( \log_b(AB) = \log_b A + \log_b B \)
• The quotient rule: \( \log_b\left(\frac{A}{B}\right) = \log_b A - \log_b B \)
• The power rule: \( \log_b(A^n) = n \cdot \log_b A \)

These rules aid in condensing or expanding expressions, making complex calculations more manageable.