Graph each logarithmic function. y=log4(2 x+3)

Name: Problem 75, Chapter 8: Algebra 2
Uploaded: 2020-01-30T02:23:50-08:00
Duration: 4 min 27 s
Channel: Ashley High
Description: Graph each logarithmic function. y=log4(2 x+3)

Key Concepts

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and is typically written in the form y = log_b(x) where b is the base and must be positive and not equal to 1. These functions have distinctive properties such as a vertical asymptote and a domain that is limited to positive input values.

Function Transformations

Transformations involve shifting, stretching, or compressing the basic graph of a function. For logarithmic functions, a transformation means adjusting the input variable (e.g., replacing x with ax + c) which results in horizontal shifts and stretches/compressions of the graph, altering its position and shape while retaining its overall logarithmic behavior.

Domain Considerations

The domain of a logarithmic function is found by setting the argument of the logarithm greater than zero. For any function expressed as log_b(ax + c), it is essential to solve the inequality ax + c > 0 to determine the valid x-values where the function exists.

Graphical Representation of Asymptotes

Logarithmic functions generally have a vertical asymptote where the function is undefined. In transformed functions like these, the boundary determined by the condition on the input (for instance, ax + c = 0) becomes the location of the vertical asymptote, guiding the shape and limits of the graph.

Transcript

00:01 This problem is asking us to graph y equals log base 4 of 2x plus 3.

00:14 Okay, so i think it's kind of easiest if we just graph this equation right here.

00:21 Long equals, sorry, y equals log base 4 of 2x, and this plus 3 is a shift to the left of three points.

00:46 So first let's graph log base 4 of 2x and then we're going to shift it.

00:53 So in order to graph it, we need to know two things.

00:57 So we need to know that in this one, when x equals 0, when 2x equals 0, what happens? what happens when 2x equals 1? and then what happens when 2x equals our base? and then we're going to make a graph plotting my x and my y.

01:20 So when 2x equals 0, x equals 0.

01:26 So my 4 of y is going to equal 0.

01:32 So this can never happen, so that is where my asymptote is going to be.

01:39 So when x equals 0, that is my vertical asymptote.

01:44 Okay, now what's going to happen when 2x equals 1? well 2x equals 1, i divide up 2 on both sides and x equals 1 half.

01:53 So 4 to the power of y equals 1 because, well, let's do that, because we wanted to equal 2x, okay? but if x equals 1 half, we know that 4 to the y equals 1.

02:15 So that means that y has to equal 0.

02:19 So when x is 1⁄2, y is 0.

02:22 And then our last 2x equals 4, dividing off my 2 on both sides.

02:27 I know that x equals 2.

02:31 So what happens, 4 to the power of y equals 2x when x is 2, 4 to the power of 5, y equals 4.

02:42 So when x equals 2, y has to equal 1, because anything raised to 1 power is going to be itself.

02:49 So we're going to graph these few things...