What is the value of [tex]$x$[/tex] in the equation below?
[tex]$1+2 e^{x+1}=9$[/tex]
A. [tex]$x=\log 4-1$[/tex]
B. [tex]$x=\log 4$[/tex]
C. [tex]$x=\ln 4-1$[/tex]
D. [tex]$x=\ln 4$[/tex]
Answer (2)
Subtract 1 from both sides of the equation: 2 e x + 1 = 8 .
Divide both sides by 2: e x + 1 = 4 .
Take the natural logarithm of both sides: x + 1 = ln ( 4 ) .
Solve for x : x = ln ( 4 ) − 1 .
The value of x is ln 4 − 1 .
Explanation
Problem Analysis We are given the equation 1 + 2 e x + 1 = 9 and asked to find the value of x . We will isolate the exponential term and then use logarithms to solve for x .
Isolating the Exponential Term First, subtract 1 from both sides of the equation: 1 + 2 e x + 1 − 1 = 9 − 1 2 e x + 1 = 8
Further Isolation Next, divide both sides by 2: 2 2 e x + 1 = 2 8 e x + 1 = 4
Applying Natural Logarithm Now, take the natural logarithm ( ln ) of both sides: ln ( e x + 1 ) = ln ( 4 )
Simplifying the Logarithm Using the property that ln ( e a ) = a , we simplify the left side: x + 1 = ln ( 4 )
Solving for x Finally, subtract 1 from both sides to solve for x :
x = ln ( 4 ) − 1
Final Answer Therefore, the value of x is ln ( 4 ) − 1 .
Examples
Exponential equations are used in various fields such as finance, physics, and engineering. For example, calculating the growth of an investment with continuous compounding involves solving an exponential equation. Suppose you invest 1000 inana cco u n tt ha tp a ys 5 A a f t er t ye a rs i s g i v e nb y A = 1000e^{0.05t}$. If you want to know how long it will take for your investment to double, you would solve the equation 2000 = 1000 e 0.05 t for t , which involves similar steps to the problem we just solved.
The value of x in the equation 1 + 2 e x + 1 = 9 is x = ln ( 4 ) − 1 . The answer corresponds to option C. Thus, C is the chosen multiple-choice option.
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