What is the solution to $4+5 e^{x+2}=11$?
A. $x=\ln \frac{7}{5}-2$
B. $x=\ln \frac{7}{5}+2$
C. $x=\ln 35-2$
D. $x=\ln 35+2$
Answer (1)
Subtract 4 from both sides: 5 e x + 2 = 7 .
Divide by 5: e x + 2 = 5 7 .
Take the natural logarithm: x + 2 = ln ( 5 7 ) .
Solve for x : x = ln ( 5 7 ) − 2 .
x = ln 5 7 − 2
Explanation
Problem Analysis We are given the equation 4 + 5 e x + 2 = 11 and we need to find the value of x that satisfies this equation.
Isolating the Exponential Term First, we isolate the exponential term by subtracting 4 from both sides of the equation: 4 + 5 e x + 2 − 4 = 11 − 4
5 e x + 2 = 7
Further Isolation Next, we divide both sides by 5 to further isolate the exponential term: 5 5 e x + 2 = 5 7
e x + 2 = 5 7
Applying Natural Logarithm Now, we take the natural logarithm of both sides of the equation to eliminate the exponential: ln ( e x + 2 ) = ln ( 5 7 )
Using the property that ln ( e a ) = a , we get: x + 2 = ln ( 5 7 )
Solving for x Finally, we solve for x by subtracting 2 from both sides: x = ln ( 5 7 ) − 2
Final Answer Therefore, the solution to the equation is x = ln ( 5 7 ) − 2 .
Examples
Exponential equations like this one are used in various fields such as finance, physics, and engineering. For example, they can model the growth of an investment, the decay of a radioactive substance, or the cooling of an object. Understanding how to solve these equations allows us to predict future values and make informed decisions in these areas. Let's say you invested money in a bank account that compounds interest continuously. The equation to model this investment is A = P e r t , where A is the final amount, P is the principal amount, r is the interest rate, and t is the time in years. If you want to know how long it will take for your investment to double, you would need to solve an exponential equation similar to the one above.
