Chapter 1.5, Problem 9QR

(a)

To determine

To find: The points of intersection of the two curves.

Answer to Problem 9QR

The intersection point of given two curves is $\left(1.58,3\right)$.

Explanation of Solution

Given information: The equations two curves are $y=2^x$ and $y=3$.

Calculation:

Substitute $3$ for $y$ in the equation $y=2^x$ to find the value of $x$.

  $3=2^x$

Take $\log$ on both sides of the equation.

  $\begin{array}{c}\log \left(3\right)=\log \left(2^x\right)\\ \log \left(3\right)=x\log \left(2\right)\\ x=\frac{\log \left(3\right)}{\log \left(2\right)}\\ \approx 1.58\end{array}$

Substitute $1.58$ for $x$ in the equation $y=2^x$.

  $\begin{array}{c}y=2^{1.58}\\ \approx 3\end{array}$

Therefore, the intersection point is $\left(1.58,3\right)$.

(b)

To determine

To find: The points of intersection of the two curves.

Answer to Problem 9QR

There are no intersection points.

Explanation of Solution

Given information: The equations two curves are $y=2^x$ and $y=-1$.

Calculation:

Since $2^x>0$ for all real number $x$ , there are no solutions to the equation $-1=2^x$ . So, there are no intersection points.

Therefore, there are no intersection points.