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Graphing Logarithmic Functions Graph the function, not by plotting points, but by starting from the graphs in Figures 4 and 9. State the domain, range, and asymptote.$$y=\log _{3}(x-1)-2$$

domain: \((1,\infty)\) Range: \((-\infty,\infty)\), Vertical Asymptote at \(x=1\)

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 3

Logarithmic Functions

Functions

Missouri State University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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02:53

Graphing Logarithmic Funct…

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Graph the function, not by…

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so we know how to grab. We know how to graft. Log base three of X is equal to is equal to why we can convert this to exponential form by saying three to the wise equal to x And so if we create a table here when y is equal to zero, X is equal to one. When y is equal to one, X is equal to three and when y is equal to two x is equal to not well. Now we know that the inside has a minus one function. So the inside the inside is subtracting one the inside of our longer than functions. So remember, if we have if we have a log base of X minus K, we're shifting the the log function to the right to the right K unit. We're shifting this to the right. So we also know that if we have something outside so we have. If we have log of eggs minus some value of K. Well, we're shifting this. This means that we're shifting this shift down K units down K units. So let's draw. Let's draw first, the log based three back. So this would be this looks like this is Axe and this is why what we have X is equal to nine X is equal to, let's say, for example, this is three and this is this is one This is one And so our graph looks like Remember, at Y is equal to zero. X is equal to one at y is equal to y is equal to one X is equal to zero. So this is Let's say, for example, this is one and this is to and we have these points here and so our graph looks like So our graph looks like member We have this this vertical asking Toda X is equal to zero. So our graph looks something like this. So this is a rough sketching, but as you can imagine, it would it would encompass these points. And so let's see what we want to shift this shift this to the right ship this to the right one unit. So in other words, at two we would have our x intercept so and then at two and then we want to shift shift downwards of two units so at at X is equal to X is equal to two instead of zero, we would have negative too. So let's draw this now so we would have This is X And this is why Well, that X is equal to two. We have negative too. Negative too. Oh, this is Let's say, for example, this is negative one. And this is negative too. Within our graph looks something like this, right? If we if we wanted to to have the same the same asking toe well then the ass into it would occur at at X is equal to one. Right? And so is our graph. Then would look something like something like this, right? And so So now we can find the domain range. So we've sketched this graph, you know, So this isn't completely accurate, but we have used what we already know, too, to sketch this and so So from this sketch, we can quickly find the domain. So the domain here is from is from positive one to infinity. Our range is still from negative infinity to infinity and are asking Tote here is at X is equal to positive one. And so these are our solutions

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