Question

If you have ever swum in a pool and your eyes began to sting and turn red, you are aware of the effects on an incorrect pH level. The pH level measures the concentration of hydronium ions and can be modeled by the function p(t) = –log10 t. The variable t represents the amount of hydronium ions, and p(t) gives the resulting pH level.

Water at 25 degrees Celsius has a pH of 7. Anything that has a pH lower than 7 is called acidic, while pH's above 7 are basic or alkaline. Seawater has a pH just over 8, while lemonade has a pH of approximately 3.

Answer 1

Create a graph of the pH function. Locate on your graph where the pH value is 0 and where it is 1. You may need to zoom in on your graph.

Answer 2

@mathmale

Answer 3

Peter, hello again! OpenStudy has been very slow today, and so I've been unable to log on. I tried to send you a private message earlier, but doubt that it went through.

How would you like to graph that function, \[pH(t)=-\log _{10}t\] ... ???

By hand: Using a TI-84 graphing calculator? or using WolframAlpha on the 'Net?

Answer 4

using geogebra, which is the same as wolfram im guessing

Answer 5

Have you tried that? I'm not familiar with Geogebra. I've just tried graphing y=-log t on wolfram alpha and will now try the same on Geogebra. You too?

Answer 6

yes, geogebra graphs logs

Answer 7

Could you tell me how to do that on Geogebra?

Answer 8

its an app that you have to download, i don't think its on the net

Answer 9

but it works just life wolfram

Answer 10

I've been trying to download and install it, but haven't yet succeeded.

Have you any experience in graphing log functions by hand?

It's a useful skill to have.

Answer 11

Also: how familiar are you with wolframalpha.com?

Answer 12

i have not been able to do it by hand, and im pretty familiar with it

Answer 13

Would you prefer, right now, to graph y=-log t by hand, by wolframalpha, or by geogebra, or perhaps in two different ways? I'll follow your lead.

Answer 14

geogebra :)

Answer 15

I'd suggest you go ahead with that. Can you share your graph with me in some manner? Trying to download geogebra has left me saddled with pop-ads. Hope you haven't experienced the same.

Answer 16

How's it going?

Answer 17

sorry open study is really slow

Answer 18

what should i input?

Answer 19

Not being familiar with geogebra, Peter, and also not having succeeded in getting it to work for me in plotting y=-log t, I'm not prepared to tell you (I don't know). However, I do have a graph produced by wolframalpha.com and will share it with you...

Answer 20

so that is the graph for the pH level?

Answer 21

Are you able to view this graph?
If so, can you estimate the t value at which the pH is 0, and then the t value for which the pH is 1? Yes, that is the graph of pH(t) = -log t .

Answer 22

This most recent graph is a partial graph of the standard y = log x.
Multiplying the right side by (-) results in the first graph I uploaded (which represents the pH as a function of t).

Answer 23

so when the pH is 0, its 3 and the pH at 1 is 0?

Answer 24

Peter, please look at the first graph I posted. The pH is on the vertical axis, whereas t is on the horizontal one.
If you follow the graph and come to the point where the graph crosses the horiz. axis, that's where the pH = 0. would you try again to identify the t value there, please?

Answer 25

it crosses at 1

Answer 26

Yes. That was one of the goals of this homework problem: to determine the t value at which the pH is zero (0).
Now use a similar technique to determine the t value for which the pH is 1.

Answer 27

but it only crosses at one point on the line

Answer 28

the curve crosses the y-axis once and the x-axis once, yes.

You've already identified the x-intercept; it's (1,0).

Now identify the y-intercept.

Answer 29

its off the graph, but im guessing its 3

Answer 30

I'm sorry, Peter, but now realize I've mislead you without meaning to. Asking you to find the y-intercept was wrong.

Instead, look at the graph and try to determine where (on the graph) the pH is = to 1. Then estimate the value of t that makes pH = 1 .

Answer 31

the ph on the y or x axis?

Answer 32

Wish I could have labelled that graph from wolframalpha!
The pH level is represented by the vertical axis, and t by the horizontal axis. So, to answer your question, look to find where the pH = 1; find the corresponding t value.

Answer 33

about 0.5

Answer 34

OK, that's about the best we can do with this particular graph from wolframalpha.

How much of this discussion is familiar for you, and how much is not? Would you be open to discussing some of the concepts involved, or do you want to move on to the next question, or what?

Answer 35

i think i got it, all i have to write is that 1 is where the pH = 0

Answer 36

and 0.5 is where the pH equals 1, correct?

Answer 37

From the graph, yes; that'd just be an estimate.

Were we to do it analytically, the answer would be different.

I'm curious: Can you evaluate log 10? log 100? log 1? Note: If I type "log", you need to assume that the base of this log system is 10.

Answer 38

log 10 would be 1
log 100 would be 10 and so on right

Answer 39

log 10 is 1, yes.
Log 100 is not 10, however. Please think about this for a moment and then try again.
Understanding this would help you do this problem quickly, practically in your head.

Answer 40

oops is 2

Answer 41

that's better!

Answer 42

:)

Answer 43

You have the function pH(t)= -log t.
Supposing that pH(t) = 1, how would you find t?

Answer 44

Hint: write the equation 1 = -log t and try to find t.

Answer 45

sub in 1

Answer 46

How would you now get rid of that "log" operator so as to solve for t?

Answer 47

log without anything in front is log 1 right

Answer 48

Not quite, Peter; " log " is an operator and can't stand alone; log must have an input (argument).

Answer 49

i don't see one ?

Answer 50

Hint: log signifies "the logarithm to the base 10".
What function is the INVERSE of y = log x?

BRB, bathroom break.

Answer 51

write the equation 1 = -log t and try to find t.

The inverse function to y = log t is y = 10^t . Note how the bases are the same : 10.
Hope you're familiar with this; if not, we need to discuss it later.

Answer 52

If you can acept that these 2 functions are inverses of one another, then let's move on to solve the equation 1 = -log t for t:

\[10^{1}=10^{-\log t}, or (10^{1})^{-t}\]

Answer 53

Can you simplify this?

Answer 54

10^1 on the left equals what on the right?

Answer 55

im not quite sure how to do the second term

Answer 56

Sorry, I 've made a mistake...I'll draw this for y ou quickly: