Jake is looking over some data regarding the strength, measured in Pascals (Pa), of some building materials and how the strength relates to the length. The data are represented by the exponential function f(x) = 2x, where x is the length. Explain how he can convert this equation to a logarithmic function when strength Pascals.

- anonymous

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- anonymous

@jim_thompson5910

- jim_thompson5910

x is the length, what is f(x)?

- anonymous

y?

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## More answers

- jim_thompson5910

in terms of the word problem

- anonymous

the strength in terms of the length

- jim_thompson5910

yes, x = length, f(x) = strength

- jim_thompson5910

so when they say `strength is 8 Pascals.`
we can replace `f(x)` with `8`

- anonymous

8=2^x

- jim_thompson5910

now convert that to a log equation

- anonymous

Ok:) is it log 2x =8?

- jim_thompson5910

I think you mean \[\Large \log_2(x) = 8\] right? you're close but not quite there

- anonymous

yes! Something along those lines

- jim_thompson5910

you'll use the rule
\[\Large b^x = y \ \ \implies \ \ \log_b(y) = x\]

- anonymous

Ok thank-you :)!!

- anonymous

what about the 8 pascals?

- jim_thompson5910

how would you use that rule to rewrite \[\Large 2^x = 8\]

- anonymous

Look at base b^x=y as if it were 2^x=8

- anonymous

Now look at the log equation and replace each variable with the corresponding one.

- jim_thompson5910

yeah so b = 2 and y = 8

- anonymous

That's it right?

- jim_thompson5910

what log equation do you have now

- anonymous

2^x = 8 means log(base 2) 8 = x

- jim_thompson5910

good

- anonymous

That's it then?

- jim_thompson5910

yeah \[\Large 2^x = 8\] turns into \[\Large \log_2(8) = x\]

- anonymous

Thank you so much~~!!

- anonymous

I have one more?

- jim_thompson5910

go ahead

- jim_thompson5910

what are your thoughts?

- anonymous

it's suppose to be 50^x. Sorry!

- anonymous

I would tell her to convert it. I dont know which formula to use now though

- jim_thompson5910

you'd use the one I just posted. The rule going from exponential to log

- anonymous

Ok. Lemme try it

- anonymous

log 50 (17)=x

- jim_thompson5910

yes, \[\Large \log_{50}(17) = x\]

- anonymous

That's it? It was that simple?

- jim_thompson5910

here is a pic that has an alternate route
http://ashikmdigitalportfolio.weebly.com/uploads/1/9/2/5/19253115/8805020_orig.jpg?146
either method gets the same answer

- anonymous

Thank you!!!

- anonymous

Can I ask one last question? Pleasee?

- jim_thompson5910

sure

- anonymous

Shannon manages a small zoo and she has been analyzing the attendance data. Shannon finds that the number of visitors increases exponentially as the temperature increases, and this situation is represented by the function f(x) = 3x. Shannon also finds a linear equation that models the number of people who leave the park early depending on the change in temperature, and it is represented by f(x) = −x + 4. The graph of the two functions is below. Find the solution to the two functions and explain what the solution represents.

- anonymous

Ty!

- jim_thompson5910

it says `The graph of the two functions is below`
where do the two functions cross?

- anonymous

oh hold on! I can graph it

- anonymous

http://fooplot.com/#W3sidHlwZSI6MCwiZXEiOiIzXngiLCJjb2xvciI6IiMwMDAwMDAifSx7InR5cGUiOjAsImVxIjoiLXgrNCIsImNvbG9yIjoiIzAwMDAwMCJ9LHsidHlwZSI6MTAwMH1d

- jim_thompson5910

looks good

- jim_thompson5910

where do the two functions cross?

- anonymous

at (1,3)

- jim_thompson5910

what does that point mean?

- anonymous

It is the solution?

- jim_thompson5910

it's the solution to the system, yes
but what does that solution mean in terms of the word problem?

- anonymous

The number of people who leave the park early due to the temperature?

- jim_thompson5910

let's go back to each function one by one
f(x) = 3^x
what is x? what is f(x)?

- anonymous

umm the increase in temperature for f(x)?

- jim_thompson5910

for f(x) = 3^x
x = temperature
f(x) = number of visitors

- jim_thompson5910

how about g(x) = -x+4 ?

- anonymous

-x=temperature or people leaving?

- jim_thompson5910

x looks like the change in temp

- anonymous

yes

- anonymous

hmm..is that when the temperature and # of people are equal?

- anonymous

Wait! NVM haha

- jim_thompson5910

so what I'm thinking is that when x = 1, the temperature change is 1 degree
so as the temp increases by 1, the number of visitors is 3 (maybe 3 thousand or something)
also, when x = 1, the number of people who leave early is 3 (thousand?)
this problem is a bit odd

- anonymous

I know right? Ugh!!! So, would (1,3) mean that that's the temp. needed to have the most visitors?

- jim_thompson5910

maybe it's when the temp change is +1, then the number of visitors equals the number of people who leave early

- jim_thompson5910

what's strange is that I don't see how you can get more visitors growing forever if it's like 100+ degrees. If anything, the attendance would go down

- anonymous

Oh! I didn't look at it that way! That is strange.

- anonymous

It's a poorly worded question tbh

- jim_thompson5910

yeah I agree

- anonymous

i'm going to go with your answer :)

- anonymous

Thanks so much for all your help today!!! You're awesome :D

- jim_thompson5910

sure thing

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