The graph of a logarithmic function is shown below.
The Correct Answer and Explanation is:
The given graph represents a logarithmic function, which generally takes the form y = log_b(x) where b is the base of the logarithm. Based on the graph’s shape and its passage through the point (1, 0), it strongly suggests that the function follows the natural logarithm or common logarithm with a positive base.
Explanation
A logarithmic function is the inverse of an exponential function, meaning it grows slowly compared to linear or exponential functions. The key characteristics of logarithmic functions include the following:
- The function is defined for positive values of x, meaning the domain is x > 0.
- The graph passes through the point (1,0) because log_b(1) = 0 for any positive base b.
- The function increases as x increases, but the rate of growth slows down progressively.
- The vertical asymptote occurs at x = 0, meaning the function approaches negative infinity as x gets closer to zero.
In real-world applications, logarithmic functions are used in situations where rapid initial growth slows over time. Examples include measuring sound intensity in decibels, modeling population growth in constrained environments, and analyzing radioactive decay.
If the graph aligns with y = log_2(x), its behavior would reflect binary logarithms, which appear frequently in computer science for analyzing algorithm efficiency. If it corresponds to y = log_10(x), it represents common logarithms used in scientific notation and scaling. Understanding logarithmic functions helps explain natural processes and technical phenomena, making them essential in mathematics and engineering.
Would you like an analysis comparing different logarithmic bases, or perhaps an example illustrating its applications in physics or chemistry?
