Learn on PengienVision, Algebra 2Chapter 6: Exponential and Logarithmic Functions

Lesson 4: Logarithmic Functions

In this Grade 11 enVision Algebra 2 lesson, students learn to graph logarithmic functions by identifying key features such as domain, range, x-intercepts, vertical asymptotes, and end behavior, using the inverse relationship between logarithmic and exponential functions. Students also apply transformations to logarithmic functions and find the equations of inverses for both exponential and logarithmic functions by interchanging variables and rewriting in logarithmic or exponential form. The lesson connects these algebraic skills to real-world contexts, such as modeling sales revenue with a logarithmic formula.

Section 1

Properties of Logarithmic Graphs

Property

Properties of the Graph of $y = \log_a x$

When $a > 1$

Domain: $(0, \infty)$
Range: $(-\infty, \infty)$
$x$ -intercept: $(1, 0)$
Asymptote: $y$ -axis
Basic shape: increasing

When $0 < a < 1$

Domain: $(0, \infty)$
Range: $(-\infty, \infty)$
$x$ -intercept: $(1, 0)$
Asymptote: $y$ -axis
Basic shape: decreasing

Section 2

Graphs of Logarithmic Functions

Property

The graph of a logarithmic function $g(x) = \log_b x$ can be found by reflecting the graph of its inverse exponential function, $f(x) = b^x$ , across the line $y=x$ . To create a table of values for $g(x) = \log_b x$ , you can interchange the columns in a table for $f(x) = b^x$ .

Examples

Since the point $(3, 27)$ is on the graph of the exponential function $y=3^x$ , the point $(27, 3)$ must be on the graph of the logarithmic function $y=\log_3 x$ .

The graph of $y=10^x$ passes through $(0, 1)$ and $(1, 10)$ . Therefore, the graph of its inverse, $y=\log_{10} x$ , must pass through $(1, 0)$ and $(10, 1)$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Properties of Logarithmic Graphs

Property

Properties of the Graph of $y = \log_a x$

When $a > 1$

Domain: $(0, \infty)$
Range: $(-\infty, \infty)$
$x$ -intercept: $(1, 0)$
Asymptote: $y$ -axis
Basic shape: increasing

When $0 < a < 1$

Domain: $(0, \infty)$
Range: $(-\infty, \infty)$
$x$ -intercept: $(1, 0)$
Asymptote: $y$ -axis
Basic shape: decreasing

Section 2

Graphs of Logarithmic Functions

Property

Examples

Since the point $(3, 27)$ is on the graph of the exponential function $y=3^x$ , the point $(27, 3)$ must be on the graph of the logarithmic function $y=\log_3 x$ .

The graph of $y=10^x$ passes through $(0, 1)$ and $(1, 10)$ . Therefore, the graph of its inverse, $y=\log_{10} x$ , must pass through $(1, 0)$ and $(10, 1)$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Properties of Logarithmic Graphs

Property

Properties of the Graph of $y = \log_a x$

When $a > 1$

Domain: $(0, \infty)$
Range: $(-\infty, \infty)$
$x$ -intercept: $(1, 0)$
Asymptote: $y$ -axis
Basic shape: increasing

When $0 < a < 1$

Domain: $(0, \infty)$
Range: $(-\infty, \infty)$
$x$ -intercept: $(1, 0)$
Asymptote: $y$ -axis
Basic shape: decreasing

Section 2

Graphs of Logarithmic Functions

Property

Examples

Since the point $(3, 27)$ is on the graph of the exponential function $y=3^x$ , the point $(27, 3)$ must be on the graph of the logarithmic function $y=\log_3 x$ .

The graph of $y=10^x$ passes through $(0, 1)$ and $(1, 10)$ . Therefore, the graph of its inverse, $y=\log_{10} x$ , must pass through $(1, 0)$ and $(10, 1)$ .