Question 149931
*[Tex \LARGE 1-(0.5)^{\frac{n}{14}}= \frac{2.5}{5.58}] Start with the given expression.


*[Tex \LARGE 1-(0.5)^{\frac{n}{14}}= 0.448] Divide {{{2.5/5.58}}} to get {{{0.448}}}


*[Tex \LARGE -(0.5)^{\frac{n}{14}}= 0.448-1] Subtract 1 from both sides.



*[Tex \LARGE -(0.5)^{\frac{n}{14}}= -0.552] Combine like terms.



*[Tex \LARGE -(0.5)^{n}= (-0.552)^{14}] Raise both sides to the 14th power. This eliminates the "14" in the denominator of the exponent.



*[Tex \LARGE -(0.5)^{n}= 0.00024] Raise -0.552 to the 14th power to get 0.00024 



*[Tex \LARGE (0.5)^{n}= -0.00024] Divide both sides by -1.



*[Tex \LARGE \log_{10}\left((0.5)^{n}\right)= \log_{10}\left(0.00024\right)] Take the log of both sides.



*[Tex \LARGE n\log_{10}\left(0.5\right)= \log_{10}\left(0.00024\right)] Rewrite the left side using the identity  {{{log(b,(x^y))=y*log(b,(x))}}}



*[Tex \LARGE n= \frac{\log_{10}\left(0.00024\right)}{\log_{10}\left(0.5\right)}] Divide both sides by {{{log(10,(0.5))}}}.




*[Tex \LARGE n= 12.025] Evaluate {{{log(10,(0.00024))/log(10,(0.5))}}} with a calculator to get 12.025



So the approximate answer is {{{n=12.025}}}