Graph each function. ƒ(x) = log₁₀ x

Verified step by step guidance

1. Understand that the function given is \( f(x) = \log_{10} x \), which is a logarithmic function with base 10. This means it is the inverse of the exponential function \( 10^x \).
2. Identify the domain of the function. Since logarithms are only defined for positive real numbers, the domain is \( x > 0 \). This means the graph will only exist to the right of the y-axis.
3. Plot key points by choosing values of \( x \) that are powers of 10, because \( \log_{10} 10^k = k \). For example, plot points at \( (1,0) \), \( (10,1) \), and \( (0.1, -1) \).
4. Draw the vertical asymptote at \( x = 0 \) because the logarithmic function approaches negative infinity as \( x \) approaches zero from the right.
5. Sketch the curve passing through the plotted points, increasing slowly and continuously for \( x > 0 \), reflecting the logarithmic growth behavior.

Key Concepts

Logarithmic Functions

A logarithmic function is the inverse of an exponential function. For ƒ(x) = log₁₀(x), it answers the question: 'To what power must 10 be raised to get x?' Understanding this helps in interpreting the behavior and values of the function.

Domain of Logarithmic Functions

The domain of ƒ(x) = log₁₀(x) includes all positive real numbers (x > 0) because logarithms of zero or negative numbers are undefined in the real number system. Recognizing the domain is essential for correctly graphing the function.

Graphing Logarithmic Functions

Graphing involves plotting points that satisfy the function and understanding key features like the vertical asymptote at x = 0, the x-intercept at (1,0), and the increasing nature of the graph. This helps visualize how the function behaves across its domain.