Problem 30 - College Algebra

Question

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. $$\log _{2}(x)=6$$

Step-by-Step Solution

Answer

The value of \( x \) is 64.

Step 1: Understand the Logarithmic Equation

The given equation is \( \log_{2}(x) = 6 \), which is in logarithmic form. This expression means that 2 raised to the power of 6 equals \( x \).

Step 2: Convert Logarithmic to Exponential Form

Using the definition of logarithms, convert the equation \( \log_{2}(x) = 6 \) to its exponential form: \( 2^6 = x \).

Step 3: Calculate the Exponential Value

Calculate the value of \( 2^6 \). This involves multiplying 2 by itself 6 times: \( 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \).

Step 4: Write the Final Solution

From the calculation in Step 3, we find that \( x = 64 \). Thus, the value of \( x \) that satisfies the given logarithmic equation is 64.

Key Concepts

Understanding Logarithmic Equations

Logarithmic equations are fascinating expressions that relate two numbers through the concept of logarithms. The basic idea is that a logarithm asks the question: "To what power must a certain base be raised, to obtain a given number?" For example, in the equation \( \log_{2}(x) = 6 \), the base is 2, and the logarithm states that 2 must be raised to the power of 6 to get \( x \).

• Logarithmic Form: \( \log_{b}(y) = x \)
• Exponential Form: \( b^{x} = y \)

The above shows how each form is structured. The logarithmic form provides a compact way to express exponential relationships, which can be very useful in solving problems involving large numbers. Learning to manipulate and convert these equations is crucial in mathematics, especially when dealing with exponential growth and decay scenarios.

Solving for x

When you are asked to solve for \( x \) in a logarithmic equation, you're finding the value that makes the equation true. In our example \( \log_{2}(x) = 6 \), solving for \( x \) requires converting the logarithmic equation into a more easily interpretable form—in this case, an exponential form.

By converting \( \log_{2}(x) = 6 \) into exponential form, you obtain \( 2^6 = x \). This step simplifies the equation, allowing you to directly compute the value of \( x \).

• Identify the base in the logarithm "2".
• Set \( x \) equal to the base raised to the power of the logarithm's result: \( 2^6 \).
• Perform the calculation: \( 2^6 = 64 \).

Thus, the solution solved via the exponential form tells us \( x = 64 \). This process illustrates how a logarithmic equation can be reformulated to find a clear answer.

Conversion Between Logarithmic and Exponential Forms

One of the fundamental skills when working with logarithmic equations is converting between logarithmic and exponential forms. Understanding how these forms relate is essential for effectively solving equations.

Let's break it down using our equation \( \log_{2}(x) = 6 \):

• Logarithmic to Exponential: Start with the logarithmic equation \( \log_{b}(y) = x \) and rewrite it as \( b^{x} = y \). In our exercise, this becomes \( 2^6 = x \).
• Exponential to Logarithmic: Conversely, if you ever need to go from exponential back to logarithmic form, simply identify the base, the exponent, and the result to form \( \log_{b}(result) = exponent \).

This conversion process is often intuitive, but practice is key. Once mastered, you can move seamlessly between these two forms and solve equations more confidently. Remember that each form provides a different perspective on the same mathematical relationship, offering flexibility in problem-solving.