The given equation is:
[tex]400\left(1.16\right)^n=35120[/tex]It is required to solve the equation for the value of n.
Divide both sides of the equation by 400:
[tex]\begin{gathered} \frac{400\left(1.16\right)^n}{400}=\frac{35120}{400} \\ \\ \Rightarrow\left(1.16\right)^n=\frac{439}{5} \end{gathered}[/tex]Take the logarithm of both sides of the equation:
[tex]\begin{gathered} \log(1.16)^n=\log\left(\frac{439}{5}\right) \\ \text{ Apply the power property of logarithms:} \\ \Rightarrow n\log(1.16)=\log\left(\frac{439}{5}\right) \end{gathered}[/tex]Divide both sides by log (1.16):
[tex]\begin{gathered} \frac{n\log(1.16)}{\log(1.16)}=\frac{\log\left(\frac{439}{5}\right)}{\log(1.16)} \\ \Rightarrow n=\frac{\operatorname{\log}(\frac{439}{5})}{\operatorname{\log}(1.16)}\approx30.151 \end{gathered}[/tex]The value of n is about 30.151.
