Question

indicate which of the following four graphs is the correct graph of this function.
$m(x) = \log_{5}(x) + 2$

Explanation:

Step1: Analyze the parent function

The parent function is \( y = \log_5(x) \), which has a vertical asymptote at \( x = 0 \) (the y - axis), a domain of \( x>0 \), and passes through \( (1,0) \) since \( \log_5(1)=0 \).

Step2: Analyze the transformation

The function \( m(x)=\log_5(x)+2 \) is a vertical shift of the parent function \( y = \log_5(x) \) up by 2 units. So the vertical asymptote remains \( x = 0 \), the domain is still \( x>0 \), and the point \( (1,0) \) on the parent function will shift to \( (1,0 + 2)=(1,2) \).

Step3: Analyze the graphs

• The first graph: The curve is below the x - axis near \( x = 1 \), which does not match the shifted function (should pass through \( (1,2) \)).
• The second graph: The curve is on the left of the y - axis (domain \( x<0 \) for part of it), but our function has domain \( x>0 \), so it is incorrect.
• The third graph: The curve passes through the origin, but our function should pass through \( (1,2) \), so it is incorrect.
• The fourth graph: It has a vertical asymptote at \( x = 0 \) (domain \( x>0 \)), and as \( x = 1 \), we can check the y - value. Since \( m(1)=\log_5(1)+2=0 + 2 = 2 \), the curve is in the region where for \( x = 1 \), \( y=2 \) (which is above the x - axis), and it has the shape of a logarithmic curve shifted up.

Answer:

The fourth graph (the bottom - right graph)