Question

Help with graphing logarithmic functions?

Answer 1

\[\Large f(x) = \frac{ 1 }{ 2 }\log_{5}(x - 3) - 5\]

Answer 2

The domain for this function is 3 < x < infinity because you cannot take the logarithm of a negative number or 0. So x - 3 > 0 or x > 3.

What is the x-intercept?

Answer 3

I need to just choose a number for the x intercept right?

Answer 4

I did the calculations. Here it happens to be a very large number and so we don't have to worry about it.

Answer 5

start with a basic f(x) = log(x) curve.
This curve lies to the right of the y-axis because x always has to be positive.
The y-axis is the vertical asymptote. As x increases, log(x) also increases. When x approaches infinity so does f(x). log(x) cuts the x-axis at x = 1 when y becomes 0.

So you can pretty much plot f(x) = log(x)

The function that is given to you is just a translation and stretching/compression of log(x)

Answer 6

Can you show me an example or something? You don't have to use my problem, but maybe a similar one so I can see it played out?

Answer 7

The drawing tool here is not too helpful in drawing curves. But you can use a free online graphing calculator. If you want ot know right away what the f(x) that is given to you looks like then you can go to https://www.desmos.com/calculator type in:
1/2log(x-3) - 5 and it will graph it for you.

But I would first start with log(x) and see what the graph looks like.
Then change log(x) to log(x-3). That will shift the log(x) curve to the LEFT by 3 units.
Then change f(x) to log(x-3) - 5 and that will shift the curve DOWN by 5 units
Then change f(x) to 1/2log(x-3) - 5 This will compress the graph vertically.

Answer 8

I need to plug in numbers for x though correct?

Answer 9

If you are going to plot this on a graph paper, then you will have to choose some x values and compute the y values. But if you want to know the general shape of f(x) the online graphing calculator will help you. Also if you click on the curve and hold the click and move the mouse the online graphing calculator will tell you what the x,y values are.

Answer 10

The only hassle IDK if there is a way to enter the logarithmic base of 5. Just log will be assumed to be log to the base 10 and so the x,y values may not apply to base 5. But the shape of the curve will be the same.

Answer 11

The graph should not go to the left of x = 3 because we established before x > 3 since we cannot take log of a negative or 0.

Answer 12

Mmk, I have the basic form from my book, I'll have to get some numbers then..do I need to use fractions or can I just use whole numbers?

Answer 13

pick x = 4, then f(4) = 1/2log(4-3) -5 = -5 because log(1) = 0
pick x = 8 next. f(8) = 1/2log_5 (8-3) - 5 = 1/2 - 5 = -4.5 (log_5 (5) = 1. that is log of 5 to the base 5 is 1).
choose x value that will make it easy for you to compute log to the base 5.
you can choose fractions such as 5^-1, 5^-2 etc for the (x-3) part so you can calculate the log to base 5 easier.

Answer 14

I don't want to choose fractions, they seem to make it more difficult haha, do I have to choose them?

Answer 15

Not necessary. If you will be using a calculator then you can plug any value of x > 3, find y, obtain a bunch of (x,y) points, choose a suitable scale for the x and y-axes and plot away.

Answer 16

By the way, if you want to get rid of the log to base 5 and replace it with log to the base 10 for easy calculations you can do so. The log will just get multiplied by a constant and the f(x) will effectively be the same.

Answer 17

\[\Large \log_{5} (M) = \log_{10} (M) * \log_{5} (10)\]

Answer 18

The last term is just a constant.

Answer 19

does your calculator allow you to choose the base of the logarithm?

Answer 20

Yes

Answer 21

ok. then set the base of log to 5, plug in a series of x values and find the y's.

Answer 22

I have bbl