Solve each equation. Give solutions in exact form. log(2 - x) = 0.5

Verified step by step guidance

1. Recall the definition of the logarithm: if \(\log_b A = C\), then \(A = b^C\). Here, the logarithm is base 10 (common logarithm), so rewrite the equation \(\log(2 - x) = 0.5\) as \$2 - x = 10^{0.5}$.
2. Express \$10^{0.5}$ as \(\sqrt{10}\) to keep the solution in exact form, so the equation becomes \(2 - x = \sqrt{10}\).
3. Isolate the variable \(x\) by subtracting \(\sqrt{10}\) from both sides: \(2 - \sqrt{10} = x\).
4. Rewrite the solution explicitly as \(x = 2 - \sqrt{10}\) to clearly state the exact form of the solution.
5. Check the solution by substituting \(x\) back into the original logarithmic expression to ensure the argument of the log, \$2 - x$, is positive, confirming the solution is valid.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential for solving log equations. For example, knowing that log_b(a) = c means b^c = a allows you to rewrite logarithmic equations in exponential form, which simplifies solving for the variable.

Domain Restrictions of Logarithmic Functions

Logarithmic functions are only defined for positive arguments. This means the expression inside the log, such as (2 - x), must be greater than zero. Identifying and applying these domain restrictions ensures that solutions are valid.

Solving Exponential Equations

After rewriting a logarithmic equation in exponential form, solving for the variable often involves isolating the variable in an exponential equation. This may require algebraic manipulation such as addition, subtraction, or division to find the exact solution.