The given equation is
[tex]4(5+8^{x+1})=100[/tex]Divide both sides by 4
[tex]\begin{gathered} \frac{4(5+8^{x+1})}{4}=\frac{100}{4} \\ \\ 5+8^{x+1}=25 \end{gathered}[/tex]Subtract 5 from each side
[tex]\begin{gathered} 5-5+8^{x+1}=25-5 \\ \\ 8^{x+1}=20 \end{gathered}[/tex]Insert log on both sides
[tex]log(8^{x+1})=log20[/tex]Use the rule of the power
[tex](x+1)log(8)=log(20)[/tex]Divide both sides by log(8)
[tex]\begin{gathered} \frac{(x+1)log(8)}{log(8)}=\frac{log(20)}{log(8)} \\ \\ x+1=\frac{log(20)}{log(8)} \end{gathered}[/tex]Subtract 1 from both sides
[tex]\begin{gathered} x+1-1=\frac{log(20)}{log(8)}-1 \\ \\ x=\frac{log(20)}{log(8)}-1 \end{gathered}[/tex]a)
The exact solution is
[tex]x=\frac{log(20)}{log(8)}-1[/tex]b)
By using the calculator the solution is
[tex]x=0.440643[/tex]The answer is x = 0.440643 to the nearest 6 decimal place
