We need to determine which of the given statements about transformed logarithmic functions is true.

1. Problem Description

We need to determine which of the given statements about transformed logarithmic functions is true.

2. Solution Steps

* Statement 1: "The range of a transformed logarithmic function is always ${y \in R}$."

The range of a standard logarithmic function $y = log_b(x)$ is indeed all real numbers. Transformations like vertical stretches, compressions, or reflections do not change the range. Vertical translations also do not change the range. Thus, the range of a transformed logarithmic function is always ${y \in R}$.

* Statement 2: "A transformed logarithmic function always has a horizontal asymptote."

Logarithmic functions have vertical asymptotes, not horizontal asymptotes. Exponential functions have horizontal asymptotes, which are inverse of logarithms. Thus, this statement is false.

* Statement 3: "The domain of a transformed logarithmic function is always ${x \in R}$."

The domain of a standard logarithmic function $y = log_b(x)$ is ${x > 0}$. Transformations like horizontal shifts can change the domain. For example, $log_b(x-1)$ has a domain of ${x > 1}$. Thus, the domain is not always ${x \in R}$. So, this statement is false.

* Statement 4: "The vertical asymptote changes when a vertical translation is applied."

A vertical translation shifts the entire graph up or down. However, it does not affect the location of the vertical asymptote. The vertical asymptote is affected by horizontal translations and stretches/compressions. Thus, this statement is false.

The only true statement is the first one.

3. Final Answer

The range of a transformed logarithmic function is always ${y \in R}$.