Solve the exponential equation using the method of relating the bases by first rewriting the equation in the form [tex]$e^{ u }=e^{ v }$[/tex].
[tex]$e^{3 x+1}=\sqrt[4]{e}$[/tex]
[tex]$x=$[/tex] (Simplify your answer. Use a comma to separate answers as needed.)
Answer (2)
Rewrite the equation in the form e 3 x + 1 = e 4 1 .
Equate the exponents: 3 x + 1 = 4 1 .
Solve for x : 3 x = 4 1 − 1 = − 4 3 .
Divide by 3 to get the final answer: x = − 4 1 .
− 4 1
Explanation
Rewrite the equation We are given the equation e 3 x + 1 = 4 e . Our goal is to solve for x . We will use the method of relating the bases by rewriting the equation in the form e u = e v .
Express both sides with the same base First, we rewrite the right side of the equation using the properties of exponents. Recall that n a = a n 1 . Therefore, 4 e = e 4 1 . So, the equation becomes e 3 x + 1 = e 4 1 .
Equate the exponents Now that we have the equation in the form e u = e v , we can equate the exponents. This gives us 3 x + 1 = 4 1 .
Isolate the term with x Next, we solve the linear equation for x . Subtract 1 from both sides: 3 x = 4 1 − 1 .
Simplify the equation Simplify the right side: 3 x = 4 1 − 4 4 = − 4 3 .
Solve for x Divide both sides by 3: x = − 4 3 ÷ 3 .
Final solution Simplify the expression for x : x = − 4 3 ⋅ 3 1 = − 4 1 . Therefore, the solution is x = − 4 1 .
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. In finance, it helps calculate investment growth, while in physics, it aids in determining the remaining amount of a radioactive substance after a certain period.
To solve the equation e 3 x + 1 = 4 e , we rewrite it as e 3 x + 1 = e 4 1 and equate the exponents. This leads to the equation 3 x + 1 = 4 1 , which simplifies to give the final answer x = − 4 1 .
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