Chapter 0.5, Problem 33E

To determine

To solve: An equation algebrically and verify solution graphically.

Answer to Problem 33E

The required algebraic solution is $t=15.744$ , and graphical $t=15.747$ .

Explanation of Solution

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

Formula used:

The log of a number $a^m$ to the base 10 is $\log \left(a^m\right)=m\log \left(a\right)$ .

Calculation:

The given equation is ${\left(1.045\right)}^t=2$ .Taking log of both sides,

  $\left.\begin{array}{c}\log {\left(1.045\right)}^t=\log 2\\ t\cdot \log \left(1.045\right)=\log 2\end{array}\right\}$ … (1)

Since, $\log \left(1.045\right)=0.01912$ and $\log 2=0.30103$ , so substitute $0.01912$ for $\log \left(1.045\right)$ and $0.30103$ for $log2$ in the result (1),

  $\begin{array}{c}t\cdot \left(0.01912\right)=0.30103\\ t=\frac{0.30103}{0.01912}\\ t=15.744\end{array}$

Write given equation in functional form as,

  $y\left(t\right)={\left(1.045\right)}^t-2$

The solution $y\left(t\right)=0$ is the solution of the equation ${\left(1.045\right)}^t-2=0$ or ${\left(1.045\right)}^t=2$ .

The graph of the function $y\left(t\right)={\left(1.045\right)}^t-2$ is shown in Fig. 1.

The curve $y\left(t\right)={\left(1.045\right)}^t-2$ intersect t-axis at point $t=15.747$ , $y\left(t\right)=0$ .

  

Hence, $t=15.747$ is the solution of the equation ${\left(1.045\right)}^t=2$ .

Thus, the algebraic solution of the equation ${\left(1.045\right)}^t=2$ is $t=15.744$ , and the graphical solution $t=15.747$ .

Therefore, the solution of the equation ${\left(1.045\right)}^t=2$ is $t=15.74$ ,