Graph of y = log(x)

Definition

The graph of y = log(x) represents the logarithmic function, where y is the logarithm of x to a specified base, typically base 10 or base e (natural logarithm). This graph showcases how logarithmic functions behave, illustrating key features like their asymptotic nature, domain, and range, and provides a visual representation of exponential growth in reverse. Understanding this graph is crucial for comprehending the relationships between exponential and logarithmic functions.

5 Must Know Facts For Your Next Test

1. The graph of y = log(x) passes through the point (1, 0) because log(1) = 0 for any base.
2. The domain of the graph is x > 0, meaning the graph does not exist for non-positive values of x.
3. The range of the graph is all real numbers (-∞, ∞), indicating that as x increases, y can take on any value.
4. As x approaches 0 from the right, the value of y approaches negative infinity, demonstrating the vertical asymptote at x = 0.
5. The slope of the graph decreases as x increases; it is steep for small values of x and gradually flattens out as x gets larger.

Review Questions

• How does the graph of y = log(x) illustrate the relationship between logarithmic and exponential functions?
  • The graph of y = log(x) demonstrates this relationship by showing how logarithmic values correspond to exponential growth. For instance, if we have y = log_b(x), it implies that b^y = x. Thus, as we plot points on the logarithmic curve, we can see that for each point on the graph, there is a corresponding exponential point that reflects growth in reverse. This connection helps reinforce how these two types of functions interact.
• What characteristics of the graph of y = log(x) indicate its asymptotic nature, and why are these important?
  • The graph of y = log(x) has a vertical asymptote at x = 0, which means it approaches but never touches or crosses this line. As x gets closer to 0 from the right, the value of y decreases without bound towards negative infinity. This behavior is crucial because it highlights the limitations of logarithmic functions and illustrates that they can only operate on positive values, which plays an essential role in various applications across mathematics and science.
• Evaluate how understanding the graph of y = log(x) can impact real-world applications in fields such as science or finance.
  • Understanding the graph of y = log(x) allows for deeper insights into phenomena such as population growth or financial modeling where exponential growth occurs. In science, this might apply to understanding decay processes or pH levels in chemistry, where logarithmic scales are used for measurements. In finance, recognizing how investments grow can help in making informed decisions based on projected growth rates. Overall, knowing how to interpret this graph translates theoretical mathematics into practical applications across various fields.

Related terms

Exponential Function:

A mathematical function of the form y = a * b^x, where a is a constant, b is the base (a positive real number), and x is the exponent. It represents rapid growth as x increases.

Asymptote:

A line that a graph approaches but never actually reaches. For the graph of y = log(x), the vertical line at x = 0 is a vertical asymptote.

Inverse Function:

A function that reverses the effect of another function. The logarithmic function is the inverse of the exponential function, meaning if y = log_b(x), then b^y = x.