The graph of y = log x is a logarithmic function, which is the inverse of the exponential function y = b^x. This function has a characteristic shape, including an asymptote at x = 0, a decreasing curve, and a range of y-values limited to positive numbers. Logarithmic functions are often used to model exponential growth and decay patterns found in various scientific and real-world phenomena. Understanding the graph of y = log x requires familiarity with concepts such as logarithms, exponents, and asymptotic behavior.
The Structure of the Graph of y log(x)
The graph of the function y = log(x) is a curve that has several important characteristics:
- Domain and range: The domain of the function is the set of all positive real numbers (x > 0), and the range is the set of all real numbers (y ∈ ℝ).
- Vertical asymptote: The graph has a vertical asymptote at x = 0, meaning that the graph approaches infinity as x approaches 0 from the right.
- Horizontal intercept: The graph has a horizontal intercept at (1, 0), meaning that the graph passes through the point (1, 0).
- Increasing: The graph is increasing for x > 0, meaning that as x increases, y also increases.
- Concave down: The graph is concave down for x > 0, meaning that the graph curves downward as x increases.
You can use the following table to summarize the key features of the graph of y = log(x):
| Feature | Value |
|---|---|
| Domain | x > 0 |
| Range | y ∈ ℝ |
| Vertical asymptote | x = 0 |
| Horizontal intercept | (1, 0) |
| Increasing | x > 0 |
| Concave down | x > 0 |
Question 1:
How can the graph of y = log x be identified?
Answer:
The graph of y = log x is an upward-sloping curve. It has an asymptote at x = 0, which means that the curve approaches but never touches the y-axis. The curve passes through the point (1, 0), and its slope increases continuously as x increases.
Question 2:
What are the characteristics of the domain and range of y = log x?
Answer:
The domain of y = log x is all positive real numbers. This is because the logarithm of a negative number is undefined. The range of y = log x is all real numbers. This is because the logarithm of any positive number can be any real number.
Question 3:
How does the transformation y = a + log(x – b) affect the graph of y = log x?
Answer:
The transformation y = a + log(x – b) shifts the graph of y = log x horizontally by b units and vertically by a units. For example, the graph of y = -1 + log(x – 2) is the graph of y = log x shifted 2 units to the right and 1 unit down.
Alright folks, that’s all for our little dive into the mysterious world of logarithmic graphs! Hopefully, you’ve gained a clearer understanding of the graph of y = log x and can confidently identify it in the future. Thanks for sticking with me through all the equations and axes. If you have any more graphing adventures, be sure to swing by again. Until next time, keep those numbers crunching and graphs plotting!