Question

use a graphing utility to solve the equation.
$e^{x}+ln x = 6$
the solution set is (use a comma to separate answers as needed. round to two decimal places as needed.)

Explanation:

Step1: Define two functions

Let \( y_1 = e^{x}+\ln x\) and \( y_2 = 6\).

Step2: Graph the functions

Use a graphing utility (such as Desmos, TI - 84 Plus etc.). For \(y_1 = e^{x}+\ln x\), the domain of \(y_1\) is \(x>0\) since the domain of \(y = \ln x\) is \(x>0\). The function \(y = e^{x}\) is an exponential function that is always positive and increasing for all real \(x\), and \(y=\ln x\) is increasing for \(x > 0\). So \(y_1=e^{x}+\ln x\) is increasing for \(x>0\).

Step3: Find the intersection point

When we graph \(y_1 = e^{x}+\ln x\) and \(y_2 = 6\) on the same set of axes (with \(x>0\)), we look for the \(x\) - value where the two graphs intersect.

Using a graphing utility (for example, on Desmos: enter \(y = e^{x}+\ln x\) and \(y = 6\) in the input bar), we find the intersection point.

Answer:

\(1.61\)