Solution:

\(\textbf{Question:}\) Find the domain and range of the logarithmic function \( f(x) = \log_{10}(x - 3) \). \[\] \(\textbf{Solution:}\)\[\] 1. **Understanding the Function:** \[ f(x) = \log_{10}(x - 3) \] In this function, the expression inside the logarithm must be greater than zero.\[\] 2. **Finding the Domain:** - For the argument \( x - 3 \) to be positive: \[ x - 3 > 0 \] Solving this inequality: \[ x > 3 \] Therefore, the domain of the function is all real numbers greater than 3. \[ \text{Domain} = (3, \infty) \] 3. **Finding the Range:** - The range of a logarithmic function \( \log_{10}(u) \) is all real numbers, because: - As the argument \( u \) approaches zero from the positive side, \( \log_{10}(u) \) approaches negative infinity. - As \( u \) increases, \( \log_{10}(u) \) also increases. - Since \( u = x - 3 \) can take any positive value: \[ \text{Range} = (-\infty, \infty) \] 4. **Summary:** - The domain of \( f(x) = \log_{10}(x - 3) \) is: \[ (3, \infty) \] - The range is: \[ (-\infty, \infty) \]