Answer: See below
Step-by-step explanation:
[tex]100\left(\frac{1}{2}\right)^{\frac{t}{214}}=20[/tex]
[tex]\mathrm{Divide\:both\:sides\:by\:}100[/tex]
[tex]\frac{100\left(\frac{1}{2}\right)^{\frac{t}{214}}}{100}=\frac{20}{100}[/tex]
[tex]Simplify[/tex]
[tex]\left(\frac{1}{2}\right)^{\frac{t}{214}}=\frac{1}{5}[/tex]
[tex]\mathrm{If\:}f\left(x\right)=g\left(x\right)\mathrm{,\:then\:}\ln \left(f\left(x\right)\right)=\ln \left(g\left(x\right)\right)[/tex]
[tex]\ln \left(\left(\frac{1}{2}\right)^{\frac{t}{214}}\right)=\ln \left(\frac{1}{5}\right)[/tex]
[tex]\mathrm{Apply\:log\:rule\:}\log _a\left(x^b\right)=b\cdot \log _a\left(x\right)[/tex]
[tex]\ln \left(\left(\frac{1}{2}\right)^{\frac{t}{214}}\right)=\frac{t}{214}\ln \left(\frac{1}{2}\right)[/tex]
[tex]\frac{t}{214}\ln \left(\frac{1}{2}\right)=\ln \left(\frac{1}{5}\right)[/tex]
[tex]\frac{\ln \left(\frac{1}{2}\right)t}{214}=\ln \left(\frac{1}{5}\right)[/tex]
[tex]\mathrm{Multiply\:both\:sides\:by\:}214[/tex]
[tex]\frac{214\ln \left(\frac{1}{2}\right)t}{214}=214\ln \left(\frac{1}{5}\right)[/tex]
[tex]-\ln \left(2\right)t=-214\ln \left(5\right)[/tex]
[tex]\mathrm{Divide\:both\:sides\:by\:}-\ln \left(2\right)[/tex]
[tex]\frac{-\ln \left(2\right)t}{-\ln \left(2\right)}=\frac{-214\ln \left(5\right)}{-\ln \left(2\right)}[/tex]
[tex]t=\frac{214\ln \left(5\right)}{\ln \left(2\right)}[/tex] or [tex]\mathrm{Decimal}:\quad t=496.89261\dots[/tex]
Answer:
[tex]\sf t= 496.9[/tex]
solving steps:
[tex]\rightarrow \sf 20=\:100\left(\dfrac{1}{2}\right)^{\dfrac{t}{214}}[/tex]
[tex]\sf \bold{divide \ both \ side \ by \ 100}[/tex]
[tex]\hookrightarrow \sf \dfrac{1}{5} =\:\left(\dfrac{1}{2}\right)^{\dfrac{t}{214}}[/tex]
[tex]\sf \bold{apply \ exponent \ rules}[/tex]
[tex]\hookrightarrow \sf ln(\dfrac{1}{5}) = ln(\:\left(\dfrac{1}{2}\right)^{\dfrac{t}{214}})[/tex]
[tex]\sf \bold{simplify}[/tex]
[tex]\hookrightarrow \sf ln(\dfrac{1}{5}) = \dfrac{t}{214} ln\:\left(\dfrac{1}{2}\right)}[/tex]
[tex]\hookrightarrow \sf \dfrac{ln(\dfrac{1}{5}) }{ln(\dfrac{1}{2})\right)}} = \dfrac{t}{214}[/tex]
[tex]\hookrightarrow \sf \dfrac{214 \ ( \ ln(\dfrac{1}{5}) \ ) }{ln(\dfrac{1}{2})\right)}} = {t}[/tex]
[tex]\hookrightarrow \sf t= 496.9[/tex]
