Graph each logarithmic function. y=log7(x-2)

Name: Problem 39, Chapter 8: Algebra 2
Uploaded: 2020-01-28T03:25:10-08:00
Duration: 3 min 48 s
Channel: Ashley High
Description: Graph each logarithmic function. y=log7(x-2)

Key Concepts

Vertical Asymptote

The vertical asymptote of a logarithmic function is the line that the graph approaches but never touches, marking the boundary of the domain. When the function’s argument is translated, the vertical asymptote shifts accordingly and represents the value of x where the logarithm's argument becomes zero.

Horizontal Translation (Shift)

A horizontal translation occurs when the input variable inside the function is adjusted by addition or subtraction. In a logarithmic function, replacing x with x minus a constant shifts the graph horizontally. This type of transformation moves the entire graph left or right without changing its overall shape.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions, meaning they answer the question of what exponent is needed to reach a given number. They have characteristic shapes, are defined only for positive arguments, and exhibit steady growth (or decay) depending on their base, making them essential in various real-world applications like measuring pH or the Richter scale.

Domain of Logarithmic Functions

The domain of a logarithmic function consists of all positive values for its argument because logarithms of zero or negative numbers are undefined. This requires setting the argument of the logarithm greater than zero, which directly impacts the range of valid x-values that can be used in the function.

Transcript

00:01 We're going to graph y equals log base 7 x minus 2.

00:11 So to do this, we are going to, so in order to find out what numbers that we want to use for a graph, we want to figure out what happens when this right here are x minus 2.

00:27 We want to know what happens when x minus 2 equals 0.

00:32 What happens when x minus 2 equals 1? 1, then what happens when x minus 2 equals our base? okay, so we're going to figure out what our xs are going to be in each of these.

00:47 So on this first one, i have to add two to both sides.

00:52 So this one, x equals 2.

00:53 So that's going to be one of our ones.

00:55 And then same thing we're going to add 2 here, x equals 3, and then add 2 here, and x equals 9.

01:06 So those are the three things that we're going to find out.

01:10 So when x equals 2, 3, and 9.

01:13 So first we're going to change this, our expression in log form.

01:20 So this now changes to 7 to the power of y equals x minus 2.

01:28 Okay, so we want to figure out what happens when what happens to our y when x equals 2, 3, and 9.

01:35 Okay, so y of 7 when if x equals 2 then this equals 0.

01:42 So that is our ascentope because there's no number that you can put in for y that's going to give you 0.

01:49 So that is our vertical asymptote.