Solve the equation.
[tex]5(e^x+2)=30[/tex]
[tex]x=[/tex] $\square$ (Round to three decimal places as needed.)
Answer (2)
To solve the equation 5 ( e x + 2 ) = 30 , we simplify it to e x = 4 by dividing and subtracting. Taking the natural logarithm gives x = ln ( 4 ) , which results in approximately 1.386 when rounded to three decimal places.
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Divide both sides of the equation by 5: e x + 2 = 6 .
Subtract 2 from both sides: e x = 4 .
Take the natural logarithm of both sides: x = ln ( 4 ) .
Calculate the value of ln ( 4 ) and round to three decimal places: x ≈ 1.386 .
Explanation
Problem Analysis We are given the equation 5 ( e x + 2 ) = 30 and we want to solve for x . We need to round the answer to three decimal places.
Divide by 5 First, divide both sides of the equation by 5: 5 5 ( e x + 2 ) = 5 30 e x + 2 = 6
Subtract 2 Next, subtract 2 from both sides: e x + 2 − 2 = 6 − 2 e x = 4
Take the Natural Logarithm Now, take the natural logarithm of both sides: ln ( e x ) = ln ( 4 ) x = ln ( 4 )
Calculate and Round Finally, calculate the value of ln ( 4 ) and round to three decimal places. The result of this operation is approximately 1.386. Therefore, x ≈ 1.386 .
Final Answer Therefore, the solution to the equation 5 ( e x + 2 ) = 30 rounded to three decimal places is x = 1.386 .
Examples
Exponential equations are used in various fields such as finance, physics, and engineering. For example, they can model population growth, radioactive decay, and compound interest. Understanding how to solve exponential equations allows us to predict future values and make informed decisions in these areas. Imagine you invest money in a bank account with continuously compounded interest. The equation to calculate the future value of your investment involves an exponential term, and solving such equations helps you determine how long it will take for your investment to reach a specific target amount.
