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Chapter 3: Problem 52

Use the One-to-One Property to solve the equation for \(x\). $$e^{2 x-1}=e^{4}$$

Short Answer

Expert verified
The solution to the equation \(e^{2x-1}=e^4\) is \(x = 5/2\).

Step by step solution

01

Applying One-to-One Property

Seeing as both sides of the equation have the same base, this is a case where the One-to-One Property can be applied. The One-to-One Property states that if \(a^m = a^n\) where a is a positive real number and a 鈮� 1, then \(m = n\). Therefore, we can set the exponents equal to each other. Hence, the equation modifies into: \(2x - 1 = 4\).
02

Solving the Equation

The next step is to solve the resulting equation for \(x\). This can be done using simple algebra. First, we will add 1 to both sides of the equation, resulting into: \(2x = 5\). Then, we divide through by 2 to get \(x = 5/2\).

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