Describe the general shape and key features of the graph of a logarithmic function, including any asymptotes.

The graph of a logarithmic function, typically expressed in the form \( y = \log_b(x) \) where \( b > 0 \) and \( b \neq 1 \), has several key features and a distinct shape: 1. **Shape**: The graph of a logarithmic function is a continuous curve that increases slowly as \( x \) increases. It starts from negative infinity and rises gradually, never reaching a maximum value. 2. **Domain**: The domain of the logarithmic function is \( (0, \infty) \). This means that the function is defined only for positive values of \( x \). 3. **Range**: The range of the logarithmic function is \( (-\infty, \infty) \). This indicates that the output can take any real number value. 4. **Intercept**: The graph intersects the y-axis at the point \( (1, 0) \) because \( \log_b(1) = 0 \) for any base \( b \). 5. **Asymptote**: The graph has a vertical asymptote at \( x = 0 \). As \( x \) approaches 0 from the right, the value of \( y \) approaches negative infinity. This means that the graph gets closer and closer to the y-axis but never touches or crosses it. 6. **Behavior**: As \( x \) increases, the function increases without bound, but the rate of increase slows down. For large values of \( x \), the logarithmic function grows very slowly compared to polynomial or exponential functions. 7. **Key Points**: Some key points on the graph include: - \( (1, 0) \) since \( \log_b(1) = 0 \) - \( (b, 1) \) since \( \log_b(b) = 1 \) - \( (b^2, 2) \) since \( \log_b(b^2) = 2 \), and so on. In summary, the graph of a logarithmic function is characterized by its gradual increase, vertical asymptote at \( x = 0 \), and the ability to take on all real values as \( x \) varies over its domain.