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Solve the following logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expression. Give the exact answer.



$\log _2 x=2$



Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

AThe solution set is { □ }. (Simplify your answer.)
BThere are infinitely many solutions.
CThere is no solution.

Verified Answer :

  • Recognize that the equation \(\log_2 x = 2\) is defined for \(x > 0\).
  • Convert the equation to exponential form \(x = 2^2\) based on the definition of a logarithm.
  • Calculate the result: \(x = 4\).
  • State the final solution set as \(\boxed{\{4\}}\).

Explanation

  1. Analysis of the Logarithmic Equation

The equation given is \(\log_2 x = 2\). The definition of a logarithm states that \(\log_b a = c\) is equivalent to the exponential form \(a = b^c\). For the expression \(\log_2 x\) to be defined, the argument \(x\) must be a positive real number (\(x > 0\)). This is because the base of the logarithm is \(2\), which is a positive number other than \(1\), and the output of the logarithm is a power to which the base must be raised to obtain the argument.

  1. Solving the Equation

To solve the equation \(\log_2 x = 2\), we apply the definition of the logarithm. By rewriting the equation in its exponential form, we have:
\[ x = 2^2 \]
Calculating the value of the exponent:
\[ x = 4 \]
We must verify that this result lies within the domain of the logarithm, which requires \(x > 0\). Since \(4 > 0\), the solution is valid.

  1. Conclusion and Final Result

Having found that \(x = 4\) is the unique solution to the equation \(\log_2 x = 2\), we conclude that the solution set consists of the single element \(\{4\}\). This corresponds to option A provided in the problem description. Therefore, the solution set is \(\{4\}\).

Examples

Logarithmic equations are essential in various fields, such as measuring the magnitude of earthquakes using the Richter scale or determining the pH level of a substance in chemistry. For instance, if you are analyzing sound intensity, you might use logarithms to convert raw power ratios into decibels, which represent how humans perceive loudness. Understanding how to solve for the unknown in a logarithmic equation allows you to reverse these processes to find original intensity levels, effectively helping you interpret logarithmic data in real-world applications.

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