Solve for t
t=-30\log_{\frac{15}{22}}\left(\frac{11}{2}\right)\approx 133.533883429
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75+110\times \left(\frac{15}{22}\right)^{\frac{t}{30}}=95
Swap sides so that all variable terms are on the left hand side.
110\times \left(\frac{15}{22}\right)^{\frac{1}{30}t}+75=95
Use the rules of exponents and logarithms to solve the equation.
110\times \left(\frac{15}{22}\right)^{\frac{1}{30}t}=20
Subtract 75 from both sides of the equation.
\left(\frac{15}{22}\right)^{\frac{1}{30}t}=\frac{2}{11}
Divide both sides by 110.
\log(\left(\frac{15}{22}\right)^{\frac{1}{30}t})=\log(\frac{2}{11})
Take the logarithm of both sides of the equation.
\frac{1}{30}t\log(\frac{15}{22})=\log(\frac{2}{11})
The logarithm of a number raised to a power is the power times the logarithm of the number.
\frac{1}{30}t=\frac{\log(\frac{2}{11})}{\log(\frac{15}{22})}
Divide both sides by \log(\frac{15}{22}).
\frac{1}{30}t=\log_{\frac{15}{22}}\left(\frac{2}{11}\right)
By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).
t=\frac{\ln(\frac{2}{11})}{\frac{1}{30}\ln(\frac{15}{22})}
Multiply both sides by 30.
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