Problem 30 - College Algebra with Corequisite Support

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 solution is \( x = 64 \).

Step 1: Understand the Problem

We are given the logarithmic equation \( \log_{2}(x) = 6 \). Our goal is to solve for \( x \) by converting the logarithmic equation to its equivalent exponential form.

Step 2: Apply the Definition of Logarithms

Recall that the logarithm \( \log_{b}(a) = c \) can be rewritten in exponential form as \( a = b^{c} \). This means that \( x = 2^{6} \) since \( \log_{2}(x) = 6 \).

Step 3: Calculate the Exponential Value

Calculate \( 2^{6} \) by multiplying 2 by itself 6 times. So, \( 2^{6} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \).

Step 4: Verify the Solution

To ensure the solution is correct, we substitute \( x = 64 \) back into the original logarithmic equation: \( \log_{2}(64) \). Since \( 2^{6} = 64 \), \( \log_{2}(64) = 6 \), confirming that \( x = 64 \) is correct.

Key Concepts

Exponential Form

Understanding the exponential form is key when solving logarithmic equations. In simple terms, converting a logarithmic equation to exponential form allows us to solve for the unknown variable. A logarithm, like \( \log_b(a) = c \), can be transformed into the exponential form \( a = b^c \). This conversion is quite powerful because it simplifies the equation from a logarithmic state to a numerical computation, making it easier to handle.

For example, in the equation \( \log_2(x) = 6 \), converting to exponential form gives us \( x = 2^6 \). This makes our task straightforward: simply calculate \( 2^6 \) to find the value of \( x \). By transforming the original equation into an exponential format, we can directly compute the desired result.

Solving Equations

Solving equations is the art of finding what values satisfy a given condition or set of conditions, typically involving mathematical symbols. In the context of logarithmic equations, this often requires a conversion step. After converting the logarithmic equation into an exponential form, the path to solving becomes clearer.

Let's go back to our example, \( \log_2(x) = 6 \). Once converted to the exponential form, \( x = 2^6 \), you now have a straightforward calculation to perform. The equation is no longer abstract and can be resolved using basic arithmetic. Here’s how it works:

• Identify the base (2 in this case) and the exponent (6).
• Multiply the base by itself as many times as the exponent indicates: \( 2 \times 2 \times 2 \times 2 \times 2 \times 2 \).
• Arrive at the solution: 64.

By following these simple steps, you can solve for \( x \), confirming that \( x = 64 \). Logarithmic equations, once converted, often become simple arithmetic problems.

Properties of Logarithms

The properties of logarithms provide rules and shortcuts that are essential for efficiently solving logarithmic equations. These properties can be used to simplify expressions, solve equations, and understand mathematical relationships more deeply. Some basic properties include:

• Product Property: \( \log_b(mn) = \log_b(m) + \log_b(n) \).
• Quotient Property: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \).
• Power Property: \( \log_b(m^n) = n \cdot \log_b(m) \).

In our exercise, however, the use of properties is less direct since the task focuses on converting and calculating. But keep in mind, these properties offer powerful techniques when handling more complex logarithmic equations. Understanding these can significantly enhance your mathematical toolkit, making solving logarithms a more intuitive and faster process.