Question

graph the logarithmic function that models the number of years, g(x), for the number of infected trees to reach a value of x.

Explanation:

Step1: Recall Logarithmic Function Form

A general logarithmic function is \( g(x)=\log_b(x) \) (or with transformations, but let's assume a basic form first, e.g., \( g(x)=\log(x) \) with base 10 or \( \ln(x) \) with base \( e \)). The domain of a logarithmic function is \( x > 0 \), and it has a vertical asymptote at \( x = 0 \).

Step2: Identify Key Points

For \( y = \log(x) \) (base 10):

• When \( x = 1 \), \( y = 0 \) (since \( \log(1)=0 \)).
• When \( x = 10 \), \( y = 1 \); when \( x=\frac{1}{10} \), \( y=-1 \), etc. For a natural log \( y = \ln(x) \), when \( x = e\approx2.718 \), \( y = 1 \); \( x = 1 \), \( y = 0 \).

Step3: Plot the Function

1. Draw the vertical asymptote at \( x = 0 \) (dashed line, since it's not part of the graph).
2. Plot the point \( (1, 0) \) (since \( \log(1)=0 \) for any base \( b>0, b

eq1 \)).

1. For a base \( > 1 \), the function is increasing. So, as \( x \) increases from 0 to \( \infty \), \( y \) increases from \( -\infty \) to \( \infty \). For example, if we take base 10, plot \( (10, 1) \); if base \( e \), plot \( (e, 1) \approx(2.718, 1) \).
2. Connect the points smoothly, approaching the vertical asymptote as \( x \to 0^+ \) (from the right side of \( x = 0 \)) and increasing slowly for \( x > 1 \), more steeply as \( x \) grows larger (but still slower than exponential).

Answer:

To graph the logarithmic function \( g(x)=\log_b(x) \) (e.g., base 10 or \( e \)):

1. Draw vertical asymptote at \( x = 0 \).
2. Plot \( (1, 0) \).
3. For \( b > 1 \), plot another point (e.g., \( (b, 1) \)) and draw a smooth curve increasing from \( -\infty \) (near \( x = 0^+ \)) through \( (1, 0) \) to \( \infty \) as \( x \to \infty \).

(Note: Since the problem mentions "the number of years" and "infected trees", the function likely models growth/decay, so a base \( > 1 \) (growth) or \( 0 < b < 1 \) (decay) logarithmic function. The graph should follow the shape of a logarithmic curve with domain \( x > 0 \), passing through \( (1, 0) \), and asymptotic to \( x = 0 \).)