Graph each function. ƒ(x) = log￬10 x

Key Concepts

Logarithmic Functions

Logarithmic functions, such as ƒ(x) = log₁₀ x, are the inverses of exponential functions. They express the power to which a base must be raised to obtain a given number. In this case, log₁₀ x asks what exponent you need to raise 10 to in order to get x. Understanding the properties of logarithms, including their domain and range, is essential for graphing them accurately.

Graphing Techniques

Graphing techniques involve plotting points on a coordinate plane to visualize the behavior of a function. For logarithmic functions, key points include (1, 0) since log₁₀(1) = 0, and (10, 1) since log₁₀(10) = 1. Additionally, recognizing the asymptotic behavior as x approaches 0 (the graph approaches negative infinity) is crucial for accurately representing the function.

Domain and Range

The domain and range of a function define the set of possible input values (domain) and the resulting output values (range). For the function ƒ(x) = log₁₀ x, the domain is x > 0, as logarithms are undefined for non-positive values. The range is all real numbers, indicating that the output can take any value from negative to positive infinity, which is important for understanding the graph's extent.