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)

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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)

Mathematics
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50^x
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.
i saw that on yahoo answers but i dont think my teacher would like the trial and error part

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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)
and for the first one there's the equal bases part
`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 \[\large 2^x = 10 \implies x = \log_{\ 2}(10) = \frac{\log(10)}{\log(2)} \approx 3.3219\]
I used the change of base formula to get the approximate decimal form
so would i write x=log50(17)
yeah \[\LARGE x = \log_{50}(17)\] then you use the change of base formula to get the approximate value of x
0.724
Rules: \[\Large b^x = y \rightarrow x = \log_b(y)\] Change of base formula \[\Large \log_{b}\left(x\right)=\frac{\log\left(x\right)}{\log\left(b\right)}\]
I'm getting 0.724 as well
so that means \[\LARGE 50^{0.724} \approx 17\]
yay thanks! what about the second part? like if the bases were equal
if the bases are equal, then you can set the exponents equal and solve for x
thanks so much you're the best! Dan who??
example \[\Large 2^3 = 2^{x+1}\] the bases are both 2, so the exponents must be equal therefore, 3 = x+1
ohh okay i get it
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)
by " f-1(x)" you mean \(\LARGE f^{-1}(x)\) right?
yes
what does that notation mean? any ideas?
no i dont understand it
it means "inverse function of f"
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
so it can be found depending on what has been done?
so would f-1(132) be 132
-132
what undoes exponents?
LOGS
YOU TAUGHT ME THAT
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
so for what it means would i just write it means the inverse of 132? or the inverse of f of 132?
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
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
why is this important? because we can use the inverse to answer questions like "in what year will the number of shoes be 132?"
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?
yeah
yaya thankyouuuu
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jim you're so nice!
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and you're patient and good at explaining things
thanks so much
no problem

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