Question 620214
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*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 5^{\frac{x}{100}}\ =\ 2]


Take the log of both sides.  Choice of base depends on whether:


1) you want an exact answer or if you have a calculating device that can give you a log value to any base you choose in which case choose base 5


2) you definitely want a numeric approximation and you only have tables or a calculating device that gives you base 10 or base e logs in which case choose base 10 or e.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \log_b\left(5^{\frac{x}{100}}\right)\ =\ \log_b\left(2\right)]


Use *[tex \LARGE \log_b\left(x^n\right)\ =\ n\log_b(x)]


to write


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{x}{100}\log_b\left(5\right)\ =\ \log_b\left(2\right)]


Solve for *[tex \LARGE x]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{100\log_b\left(2\right)}{\log_b(5)}]


Like I said, use any base you like, but if you just want to render the exact answer use base 5 because *[tex \LARGE \log_b(b)\ =\ 1], so the expression would reduce to:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ 100\log_5\left(2\right)]



John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
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