Questions: Question 7 Find the domain and range of (y=log 6(2+5 x)). The domain is: The range is :

Solution Steps

To find the domain and range of the function \( y = \log_{6}(2 + 5x) \):

1. Domain: The argument of the logarithm (inside the log function) must be greater than zero. So, solve the inequality \( 2 + 5x > 0 \) to find the domain.
2. Range: The logarithmic function can take any real number as its output, so the range is all real numbers.

To find the domain of \( y = \log_{6}(2 + 5x) \), we need the argument of the logarithm to be greater than zero: \[ 2 + 5x > 0 \] Solving for \( x \): \[ 5x > -2 \] \[ x > -\frac{2}{5} \] Thus, the domain is: \[ x > -\frac{2}{5} \]

The range of a logarithmic function \( y = \log_{a}(x) \) for any base \( a > 1 \) is all real numbers. Therefore, the range of \( y = \log_{6}(2 + 5x) \) is: \[ (-\infty, \infty) \]

Final Answer