To find the domain and range of a logarithmic function of the form

follow these steps:

1. Domain

The domain of a logarithmic function is determined by the argument of the logarithm, $x-h$. The argument must be positive because logarithms are not defined for zero or negative values.

Rule: Solve the inequality $x-h>0$:

The domain is $x\in (h,\infty )$.

2. Range

The range of a logarithmic function is not affected by the transformations $h$ and $k$. A logarithmic function can output any real number regardless of horizontal shifts ($h$) or vertical shifts ($k$).

Rule: The range of $f(x)={\log }_b(x-h)+k$ is always:

Example:

Find the domain and range of the function

1. Domain:

• The argument of the logarithm is $x-4$.
• Solve $x-4>0$:

• The domain is $(4,\infty )$.

2. Range: The range of any logarithmic function is all real numbers, regardless of shifts. So the range is $(-\infty ,\infty )$.

Final Answer:

For $f(x)={\log }_3(x-4)+2$:

• Domain: $(4,\infty )$
• Range: $(-\infty ,\infty )$