Question

find range
$g(x) = 3\log_{2}(7x - 5) + 4$

Explanation:

Step1: Recall range of basic log function

The basic logarithmic function \( \log_b(u) \) (here \( b = 2 \)) has a range of \( (-\infty, \infty) \) for \( u>0 \).

Step2: Analyze the inner function's effect on domain (not range yet)

For \( g(x)=3\log_2(7x - 5)+4 \), the argument \( 7x - 5>0 \) (domain consideration), but for range, we look at the transformation of \( \log_2(7x - 5) \).

Step3: Analyze vertical stretch and shift

The function \( \log_2(7x - 5) \) has range \( (-\infty, \infty) \). Multiplying by 3 (a vertical stretch) still gives a range of \( (-\infty, \infty) \) because stretching a line (or the set of all real numbers) by a non - zero constant doesn't change the fact that it spans all real numbers. Then adding 4 (a vertical shift) also doesn't restrict the range. So \( 3\log_2(7x - 5) \) has range \( (-\infty, \infty) \), and \( 3\log_2(7x - 5)+4 \) also has range \( (-\infty, \infty) \).

Answer:

The range of \( g(x)=3\log_2(7x - 5)+4 \) is \( (-\infty, \infty) \) (or all real numbers \( \mathbb{R} \))