Explanation:
To solve the equation 4e^(2x) - 1 = 13 for x, we will follow these steps:
Step 1: Add 1 to both sides of the equation to isolate the term with the exponential:
4e^(2x) = 14
Step 2: Divide both sides of the equation by 4 to isolate the exponential term:
e^(2x) = 14/4
Simplifying the right side:
e^(2x) = 7/2
Step 3: Take the natural logarithm (ln) of both sides of the equation to eliminate the exponential:
ln(e^(2x)) = ln(7/2)
By the properties of logarithms, the ln and e^(2x) cancel each other out:
2x = ln(7/2)
Step 4: Divide both sides of the equation by 2 to solve for x:
x = (1/2) ln(7/2)
Thus, the solution to the equation 4e^(2x) - 1 = 13 is x = (1/2) ln(7/2).