Question

graph with points (1, 0), (3, 1), (0.33, -1). options: ( y = log_{0.4} x ), ( y = log_{1} x ), ( y = log_{3} x ), ( y = log_{10} x )

Explanation:

Step1: Recall log function properties

For a logarithmic function \( y = \log_b x \), when \( x = 1 \), \( y = 0 \) (since \( \log_b 1 = 0 \) for any \( b>0, b
eq1 \)). When \( y = 1 \), \( x = b \) (since \( \log_b b = 1 \)).

Step2: Analyze the graph

From the graph, we see the point \( (3, 1) \). So when \( y = 1 \), \( x = 3 \). Using the property \( \log_b b = 1 \), this means \( b = 3 \), so the function should be \( y=\log_3 x \). Let's check other options:

• For \( y = \log_{0.4}x \), it should be a decreasing function (since \( 0 < 0.4 < 1 \)), but the graph is increasing, so eliminate.
• \( y = \log_1 x \) is undefined (base of log can't be 1), eliminate.
• For \( y=\log_{10}x \), when \( y = 1 \), \( x = 10 \), but the graph has \( x = 3 \) when \( y = 1 \), so eliminate.

Answer:

\( y = \log_3 x \) (the option \( y=\log_3 x \))