Erika was working on solving the exponential equation 50x = 17; however, she is not quite sure where to start. Using complete sentences, describe to Erika how to solve this equation and how solving would be different if the bases were equal. (10 points)

- anonymous

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- anonymous

50^x

- Kash_TheSmartGuy

Erika could solve with a calculator and do trial and error until she found some power of 5 that would equal 17 (it's a little above 1.76 by this method). But being a smart person, Erika would see that the best solution would be to take logs of both sides
Log 5^x = log 17....... Erika knows that log 5^x = x(log 5)
x(log 5) = log 17
0.69897x = 1.2304489....divide across by 0.69897
x = 1.76055, a much more precise solution
With the above you can surely write a few sentences.

- anonymous

i saw that on yahoo answers but i dont think my teacher would like the trial and error part

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## More answers

- anonymous

It's ok ill use that for that question but how about this one Brett has determined a function f(x) that shows the exponential growth of the number of shoes Larae owns each year. Explain how the f-1(x) can be found and what f-1(132) means. (10 points)

- anonymous

@dan815

- anonymous

and for the first one there's the equal bases part

- anonymous

@jim_thompson5910

- jim_thompson5910

`Erika was working on solving the exponential equation` \(\LARGE 50^x = 17\) `; however, she is not quite sure where to start. Using complete sentences, describe to Erika how to solve this equation and how solving would be different if the bases were equal.`
use logs to isolate exponents
example
\[\large 2^x = 10 \implies x = \log_{\ 2}(10) = \frac{\log(10)}{\log(2)} \approx 3.3219\]

- jim_thompson5910

I used the change of base formula to get the approximate decimal form

- anonymous

so would i write x=log50(17)

- jim_thompson5910

yeah \[\LARGE x = \log_{50}(17)\]
then you use the change of base formula to get the approximate value of x

- anonymous

0.724

- jim_thompson5910

Rules:
\[\Large b^x = y \rightarrow x = \log_b(y)\]
Change of base formula
\[\Large \log_{b}\left(x\right)=\frac{\log\left(x\right)}{\log\left(b\right)}\]

- jim_thompson5910

I'm getting 0.724 as well

- jim_thompson5910

so that means \[\LARGE 50^{0.724} \approx 17\]

- anonymous

yay thanks! what about the second part? like if the bases were equal

- jim_thompson5910

if the bases are equal, then you can set the exponents equal and solve for x

- anonymous

thanks so much you're the best! Dan who??

- jim_thompson5910

example
\[\Large 2^3 = 2^{x+1}\]
the bases are both 2, so the exponents must be equal
therefore, 3 = x+1

- anonymous

ohh okay i get it

- anonymous

what about for the brett problem? Brett has determined a function f(x) that shows the exponential growth of the number of shoes Larae owns each year. Explain how the f-1(x) can be found and what f-1(132) means. (10 points)

- jim_thompson5910

by " f-1(x)" you mean \(\LARGE f^{-1}(x)\) right?

- anonymous

yes

- jim_thompson5910

what does that notation mean? any ideas?

- anonymous

no i dont understand it

- jim_thompson5910

it means "inverse function of f"

- jim_thompson5910

the inverse undoes whatever operation was applied
so say you add initially, the inverse would be subtraction
if you multiply, the inverse is division
if you square something, the inverse is the square root

- anonymous

so it can be found depending on what has been done?

- anonymous

so would f-1(132) be 132

- anonymous

-132

- jim_thompson5910

what undoes exponents?

- anonymous

LOGS

- anonymous

YOU TAUGHT ME THAT

- jim_thompson5910

yes you will use logs to get the inverse of f
we can't actually find the inverse since we don't know what the function f is

- anonymous

so for what it means would i just write it means the inverse of 132? or the inverse of f of 132?

- jim_thompson5910

the original f(x) function takes an x value, which is the number of years, and produces a y value
y = f(x)
in goes x ----> out comes y or f(x)
x = number of years
y = number of shoes

- jim_thompson5910

the inverse takes everything in reverse because we're undoing everything
with the inverse, the y value is now the input, the x is the output
in goes y into the inverse ----> out comes x

- jim_thompson5910

why is this important? because we can use the inverse to answer questions like "in what year will the number of shoes be 132?"

- anonymous

so it would be right if i wrote that f-1(132) means that you take the inverse of it and now the y value is the input and the x is the output?

- jim_thompson5910

yeah

- anonymous

yaya thankyouuuu

- jim_thompson5910

|dw:1436396943396:dw|

- anonymous

jim you're so nice!

- jim_thompson5910

|dw:1436396954380:dw|

- anonymous

and you're patient and good at explaining things

- anonymous

thanks so much

- jim_thompson5910

no problem

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