Graph Logarithmic Functions - Algebra-2

1. Fundamental Concepts

Relationship Between Logarithmic and Exponential Functions

A logarithmic function is the inverse function of an exponential function. Their graphs are symmetric about the line , with the core corresponding relationship as follows:

Type of Function	General Form	Core Connection (Mutually Inverse Functions)
Exponential Function	(where and )	If a point on the exponential function is , the corresponding point on the logarithmic function is
Logarithmic Function	$y = \log_b x$ (where $b>0$, $b≠1$, and $x>0$)	For example: The inverse function of the exponential function $y = 10^x$ is the logarithmic function $y = \log_{10} x$ (which can be abbreviated as $y = \log x$)

Domain and Range of Logarithmic Functions

Domain (D): The independent variable x of a logarithmic function must satisfy , so the domain is . This is because the result of the exponential function is always positive, and as the input of the inverse logarithmic function (which is the output of the original exponential function), it can only be a positive number.
Range (R): The function value y of a logarithmic function can take any real number, so the range is . This is because the exponent y of the exponential function can take any real number, and as the output of the inverse logarithmic function (which is the exponent of the original exponential function), it can also cover all real numbers.

Key Fixed Point of Logarithmic Functions

All logarithmic functions $y = \log_b x$ (where $b>0$ and $b≠1$) must pass through the fixed point $(1, 0)$, which is derived as follows:

When $x = 1$, according to the definition of logarithms, "$\log_b 1 = 0$" (since $b^0 = 1$). Therefore, no matter what value the base $b$ takes (as long as it meets the conditions), the graph of the function will pass through $(1, 0)$.

For example: $y = \log x$ (with base 10) passes through $(1, 0)$, and $y = \log_e x$ (natural logarithm, with base $e$) also passes through $(1, 0)$.

2. Key Concepts

Basic Characteristics of Logarithmic Function Graphs

Taking the common logarithmic function $y = \log x$ (with base $b = 10 > 1$) as an example, its graph has the following characteristics:

Monotonicity: When the base , the logarithmic function is monotonically increasing on its domain (i.e., the larger x is, the larger y is); if the base , the function is monotonically decreasing (i.e., the larger x is, the smaller y is).
Asymptote: The graph has a vertical asymptote (the y-axis). The graph of the function approaches the y-axis infinitely but never intersects it (because the domain is , so x cannot be equal to 0).
Intersection with Coordinate Axes: It only intersects the x-axis at the fixed point and has no intersection with the y-axis (since is not in the domain).

Symmetric Relationship Between Exponential and Logarithmic Function Graphs

Taking $y = 10^x$ (exponential function) and $y = \log x$ (logarithmic function) as examples:

The exponential function $y = 10^x$ passes through the fixed point $(0, 1)$, and the logarithmic function $y = \log x$ passes through the fixed point $(1, 0)$. These two points are symmetric about the line $y = x$.

For any point $(a, 10^a)$ on $y = 10^x$, its symmetric point $(10^a, a)$ about the line $y = x$ must lie on $y = \log x$. This is a core property of inverse function graphs.

3. Examples

Example

Question: Graph the function $y = \log_2(x)$ .

Step-by-Step Solution:

Identify key points: $(1, 0)$ , $(2, 1)$ , $\left(\frac{1}{2}, -1\right)$ .
Plot these points on the coordinate plane.
Draw a smooth curve passing through these points with a vertical asymptote at $x = 0$ .

Validation: Check that the graph approaches the y-axis but never touches it, and passes through the identified points.

Question: Write the domain, range, and identify the fixed point that the logarithmic function $y = \log_2 x$ must pass through.

Solution:

Domain: $x > 0$, i.e., $(0, +\infty)$;

Range: $y \in \mathbb{R}$, i.e., $(-\infty, +\infty)$;

Fixed Point: When $x = 1$, $y = \log_2 1 = 0$, so the fixed point is $(1, 0)$.

Question: Given that there is a point $P$ on the graph of the logarithmic function $y = \log_3 x$ with an abscissa of 9, find the ordinate of point $P$; if the ordinate of another point $Q$ on the function graph is -1, find the abscissa of point $Q$.

Solution:

Find the ordinate of point $P$: Given $x = 9$, substitute it into the function to get $y = \log_3 9$. Since $3^2 = 9$, $y = 2$, so the coordinates of point $P$ are $(9, 2)$;

Find the abscissa of point $Q$: Given $y = -1$, that is, $\log_3 x = -1$. According to the definition of logarithms, $3^{-1} = x$, so $x = \frac{1}{3}$, and the coordinates of point $Q$ are $(\frac{1}{3}, -1)$.

4. Problem-Solving Techniques

Visual Strategy: Use graph paper or graphing software to accurately plot points and visualize transformations.
Error-Proofing: Always check the domain restrictions and asymptotes before plotting.
Concept Reinforcement: Practice identifying the inverse relationship between exponential and logarithmic functions by graphing both on the same coordinate system.