Chapter 1.5, Problem 33E

To determine

To solve: The equation ${\left(1.045\right)}^t=2$ and support the solution graphically.

Answer to Problem 33E

The solution of the equation ${\left(1.045\right)}^t=2$ is approximately $15.75$.

Explanation of Solution

Given information: The equation is ${\left(1.045\right)}^t=2$.

Calculation:

Take the $\log$ on both sides of the equation ${\left(1.045\right)}^t=2$ and simplify it by power Rule.

  $\begin{array}{c}\log {\left(1.045\right)}^t=\log 2\\ t\log \left(1.045\right)=\log 2\\ t=\frac{\log 2}{\log \left(1.045\right)}\\ \approx 15.75\end{array}$

The solution of the equation is 15.75.

To verify the solution graphically, plot the graph of the equation ${\left(1.045\right)}^t=2$ . The intersection point of these two graphs will be the solution.

Follow the steps to solve the equation by calculator:

• First press “ON” button on graphical calculator and press $\boxed{\text{Y}=}$ key.
• Enter right hand side of the equation after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{1.045}\boxed{)}\boxed{^}\boxed{\text{X}}$ . Now, press the key $\boxed{\text{ENTER}}$ .
• Enter right left side of the equation after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{2}$ .
• Press $\boxed{\text{WINDOW}}$ key and adjust the window at $\left[-2,18\right]$ by $\left[-23\right]$ .
• Now, press keystrokes given below to find the value of $t$ as:

  $\boxed{2\text{nd}}\to \boxed{\text{TRACE}}\to \boxed{5}\to \boxed{\text{ENTER}}\to \boxed{\text{ENTER}}\to \boxed{\text{ENTER}}$

  

From the graph, it can be observed that the intersection point is $\left(15.75,2\right)$ at $y=2$.

Therefore, the solution of the equation ${\left(1.045\right)}^t=2$ is approximately $15.75$.