Question

what is the value of x in the equation below?
1 + 2e^{x + 1}=9
x = log4 - 1
x = log4
x = ln4 - 1
x = ln4

Explanation:

Step1: Isolate the exponential term

Subtract \(1\) from both sides of the equation \(1 + 2e^{x + 1}=9\).
\(2e^{x + 1}=9 - 1\)
\(2e^{x + 1}=8\)
Divide both sides by \(2\):
\(e^{x + 1}=\frac{8}{2}=4\)

Step2: Use the property of natural logarithm

Since \(y = e^{x}\) and \(x=\ln(y)\) (where \(\ln\) is the natural logarithm, i.e., \(\log_{e}\)), for the equation \(e^{x + 1}=4\), we take the natural logarithm of both sides.
\(\ln(e^{x + 1})=\ln(4)\)
Using the property \(\ln(e^{a})=a\), we get \(x + 1=\ln(4)\)
Subtract \(1\) from both sides:
\(x=\ln(4)-1\)

Answer:

\(x = \ln4-1\) (the third option)