EXPLANATION:
Given;
We are given the following equation;
[tex]1.12^x=20[/tex]Required;
We are required to describe two methods which can be used to solve for x in this equation.
Step-by-step solution;
We can solve for the variable x by taking the natural log of both sides of the equation. This is shown below;
[tex]1.12^x=20[/tex]We take the natural log of both sides;
[tex]ln1.12^x=ln20[/tex]Next we apply the log rule;
[tex]\begin{gathered} If: \\ log_bx^a \\ Then: \\ alog_bx \end{gathered}[/tex]Therefore, our equation is now refined and becomes;
[tex]xln1.12=ln20[/tex]Divide both sides by ln(1.12);
[tex]x=\frac{ln(20)}{ln(1.12)}[/tex]A second method is to express it as a logarithmic equation;
[tex]1.12^x=20[/tex]We shall apply the log rule which is;
[tex]\begin{gathered} If: \\ log_bx=a \end{gathered}[/tex][tex]\begin{gathered} Then: \\ b^a=x \end{gathered}[/tex]For example;
[tex]\begin{gathered} If: \\ log_{10}100=2 \end{gathered}[/tex][tex]\begin{gathered} Then: \\ 10^2=100 \end{gathered}[/tex]Therefore, for the equation given;
[tex]\begin{gathered} If: \\ 1.12^x=20 \end{gathered}[/tex][tex]\begin{gathered} Then: \\ log_{1.12}20=x \end{gathered}[/tex]Note that both solutions can be simplified eventually with the use of a calculator.
ANSWER:
(1) By taking the natural log of both sides
(2) By expressing the equation as a logarithmic equation